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Similar Question 1
<p>Identify a pattern and find the next three numbers in the pattern.</p><p><code class='latex inline'>\displaystyle 30,45,60,75, \ldots </code></p>
Similar Question 2
<p>The first four diagrams in a pattern show asterisks arranged in an m-shape.</p><p>a) Copy and complete the table for the four diagrams.<code class='latex inline'>\displaystyle \begin{array}{|c|c|} \hline Diagram Number, d & Number of Asterisks, a \\ \hline 1 & \\ \hline 2 & \\ \hline 3 & \\ \hline 4 & \\ \hline \end{array} </code></p><p>b) Use finite differences to determine whether the function is linear or quadratic.</p><p>c) Determine an equation of the function by inspection or by using your graphing calculator.</p><p>d) How many asterisks will there be in the 15 th diagram? the 21 st diagram?</p><p>e) Which diagram contains 131 asterisks?</p><img src="/qimages/156707" />
Similar Question 3
<p>Given the formula for the general term of an arithmetic sequence, write the first five terms. Graph the discrete function that represents each sequence.</p><p><code class='latex inline'>t_n = 4n - 1</code></p>
Similar Questions
Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>State whether or not each sequence is arithmetic. Justify your answer.</p><p><code class='latex inline'>-12, -5, 2, 9</code>, ...</p>
<p>Determine whether each sequence is geometric. If so, find the common ratio.</p><p><code class='latex inline'>\displaystyle -1,1,-1,1, \ldots </code></p>
<p>For each arithmetic sequence, determine the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>. Then, write the next four terms.</p><p>-3, 2, 7, ...</p>
<p>A number, <code class='latex inline'>m</code>, is called an arithmetic mean between <code class='latex inline'>a</code> and <code class='latex inline'>b</code> if <code class='latex inline'>a</code>, <code class='latex inline'>m</code>, and <code class='latex inline'>b</code> form an arithmetic sequence.</p><p>Determine five arithmetic means between 5 and 29.</p>
<p>The graphs show the terms in a sequence. Write each sequence in function notation and specify the domain.</p><img src="/qimages/23293" />
<p>State whether or not each sequence is arithmetic. Justify your answer.</p><p>1, 4, 7, 10, ...</p>
<p>Write a recursive definition for each sequence.</p><p><code class='latex inline'>\displaystyle 100,10,1,0.1,0.01, \ldots </code></p>
<p>Use appropriate tools and strategies to find the next three terms in each sequence.</p> <ul> <li><code class='latex inline'> \displaystyle \frac{2}{3}, \frac{7}{12}, \frac{1}{2}, \frac{5}{12} </code></li> </ul>
<p>Determine the number of terms in each geometric sequence. </p><p>2, -10, 50, ..., -156 250</p>
<p>Determine whether each sequence is geometric. If so, find the common ratio.</p><p><code class='latex inline'>\displaystyle -1,-6,-36,-216, \ldots </code></p>
<p>The graphs show the terms in a sequence. Write each sequence in function notation and specify the domain.</p><img src="/qimages/23298" />
<p>For each geometric sequence, determine the formula for the general term and use it to determine the indicated term.</p><p>13.45, 2.69, 0.538, ..., <code class='latex inline'>t_10</code></p>
<p>Given the formula for the general term of an arithmetic sequence, write the first five terms. Graph the discrete function that represents each sequence.</p><p><code class='latex inline'>f(n) = \dfrac{3n + 2}{2}</code></p>
<p>If you fold a piece of string in half, in half again, and so on, up to n folds, and then cut it through the middle with a pair of scissors, how many pieces of string will you have?</p>
<p>Determine whether each sequence is geometric. If so, find the common ratio.</p><p><code class='latex inline'>\displaystyle 10,4,1.6,0.64, \ldots </code></p>
<p>What is the common ratio in the geometric sequence <code class='latex inline'>\displaystyle \frac{9}{2}, 3,2, \frac{4}{3}, \ldots </code> ?</p><p>F) <code class='latex inline'>\displaystyle \frac{3}{2} </code></p><p>G) <code class='latex inline'>\displaystyle \frac{9}{2} </code> </p><p>(H) <code class='latex inline'>\displaystyle \frac{2}{3} </code> </p><p>D) <code class='latex inline'>\displaystyle \frac{27}{2} </code></p>
<p>Find the first six terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=\frac{1}{2} n^{3}-1 </code></p>
<p>The graphs show the terms in a sequence. Write each sequence in function notation and specify the domain.</p><img src="/qimages/23296" />
<p>Find the unknown terms, <code class='latex inline'>m</code> and <code class='latex inline'>n</code>, in each geometric sequence.</p><p><code class='latex inline'>m, n, 2, \dfrac{1}{2}</code></p>
<p>Write the 16th term, given the explicit formula for the nth term of the sequence.</p><p><code class='latex inline'>f(n) = 5 - 2n</code></p>
<p>Find the next three terms in each sequence. Describe how to find successive terms.</p><p>a) 1, 2, 4, 7</p><p>b) 1, 4, 9, 16</p><p>c) 17, 12, 7, 2</p><p>d) 2, 6, 12, 20</p>
<p>State the common ratio for each geometric sequence and write the next three terms.</p><p>8000, 800, 80, 8, ...</p>
<p>Which term of the geometric sequence <code class='latex inline'>5, 1, \dfrac{1}{5}</code>, ... has a value of <code class='latex inline'>\dfrac{1}{78125}</code>?</p>
<p>Find the next two items for each pattern. Then find the 21st figure in the pattern.</p><img src="/qimages/24677" />
<p>Investigate the point of intersection of the lines <code class='latex inline'>ax + by=c</code> and <code class='latex inline'>dx + ey=f</code>, where <code class='latex inline'>a, b, c</code> and <code class='latex inline'>d, e, f</code> are both arithmetic sequences (an arithmetic sequence is a sequence with constant first differences). Write a summary of your findings.</p>
<p>Determine the first five terms of two different geometric sequences that satisfy the given conditions.</p><p>The sum of the first two terms of the sequence is 3 and the sum of the next two terms is <code class='latex inline'>\frac{4}{3}</code>.</p>
<p>Determine whether each list is a sequence or a series and finite or infinite.</p><p><code class='latex inline'>\displaystyle 1,2,4,8,16,32, \ldots </code></p>
<p>Find the value of <code class='latex inline'>y</code> that makes <code class='latex inline'>y+8, 4y+6, 3y,...</code> an arithmetic sequence.</p>
<p>Use an appropriate tool to help determine which term in the sequence <code class='latex inline'>100, 93, 86,</code> ... is <code class='latex inline'>-600</code>.</p>
<p>The first four diagrams in a pattern show asterisks arranged in an m-shape.</p><p>a) Copy and complete the table for the four diagrams.<code class='latex inline'>\displaystyle \begin{array}{|c|c|} \hline Diagram Number, d & Number of Asterisks, a \\ \hline 1 & \\ \hline 2 & \\ \hline 3 & \\ \hline 4 & \\ \hline \end{array} </code></p><p>b) Use finite differences to determine whether the function is linear or quadratic.</p><p>c) Determine an equation of the function by inspection or by using your graphing calculator.</p><p>d) How many asterisks will there be in the 15 th diagram? the 21 st diagram?</p><p>e) Which diagram contains 131 asterisks?</p><img src="/qimages/156707" />
<p>Determine an explicit formula for the nth term of each sequence. Use the formula to write the 15th term.</p><p><code class='latex inline'>2, \dfrac{2}{\sqrt 2}, \dfrac{2}{\sqrt 3}, 1, ...</code></p>
<p>In a bacteria strain, the number of bacteria doubles every 20 min. There were 8 bacteria to start with.</p><p> How many bacteria will there be after 3 h?</p>
<p>Chad, a champion show dog, had 2 parents one generation ago, 4 grandparents two generations ago, 8 great-grandparents three generations ago, and so on.</p><p>How many generations ago did Chad have 8192 ancestors?</p>
<p>Describe the pattern in each sequence. Write the next three terms of each sequence.</p><p><code class='latex inline'>ar^3, ar^2, ar, a, ...</code></p>
<p>Determine whether each sequence is geometric. If so, find the common ratio.</p><p><code class='latex inline'>\displaystyle 7,0.7,0.07,0.007, \ldots </code></p>
<p>State the common ratio for each geometric sequence and write the next three terms.</p><p><code class='latex inline'>\dfrac{1}{2}, -1, 2, -4, ...</code></p>
<p>Eight fence posts are to be equally spaced between two corner posts that are 117 m apart.</p><p>How is the value you found in part a) related to the general term of the formula you found in part c)?</p>
<p>Given the formula for the general term of an arithmetic sequence, determine <code class='latex inline'>t_{16}</code>.</p><p><code class='latex inline'>f(n) = 19 - 12n</code></p>
<p>Determine an explicit formula for the nth term of each sequence. Use the formula to write the 15th term.</p><p>-16, 8, -4, 2, ...</p>
<p>Find the 17 th term of the arithmetic sequence.</p><p><code class='latex inline'>\displaystyle a_{18}=32, d=-4 </code></p>
<p>Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms in the sequence.</p><p><code class='latex inline'>-18, -16, -14, -12,...</code></p>
<p>Describe the pattern in the sequence. Give the next two terms.</p><img src="/qimages/152700" />
<p>Eight fence posts are to be equally spaced between two corner posts that are 117 m apart.</p><p>Write a sequence to represent this situation.</p>
<p>Describe the pattern in the sequence. Give the next two terms.</p><p>3, 3, 6, 18, 72</p>
<p>Determine whether the sequence is arithmetic, geometric, or neither. Give a reason for your answer.</p><p>7, 5, 3, 1, ...</p>
<p>Determine an explicit formula for the nth term of each sequence. Use the formula to write the 15th term.</p><p><code class='latex inline'>2, \dfrac{3}{2}, \dfrac{4}{3}, \dfrac{5}{4}, ...</code></p>
<p>For each sequence, make a table of values using the term number and term, and calculate the finite differences. Then, determine an explicit formula in function notation and specify the domain.</p><p>2, 5, 10, 17, ...</p>
<p>Find the first six terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=(-3)^{n} </code></p>
<p>Describe the pattern in the sequence. Give the next two terms.</p><p><code class='latex inline'> \displaystyle \frac{12}{5}, 2, \frac{8}{5} </code></p>
<p>Eight fence posts are to be equally spaced between two corner posts that are 117 m apart.</p><p>Write the formula for the general term of the sequence in part b).</p>
<p>Determine the number of terms in each geometric sequence. </p><p><code class='latex inline'>a, ab, ab^{2}, ..., ab^{12}</code></p>
<ol> <li>Square pattern The first five diagrams in a pattern are made up of squares of side length 1 unit.</li> </ol> <p>a) Copy and complete the tables for the five diagrams.</p><p>b) Use finite differences to determine whether the function shown in each table is linear or quadratic.</p><p>c) Determine an equation of each function by using your graphing calculator.</p><p>d) Find the perimeter and the area of the diagram of base length 15 units; of base length 30 units.</p><p>Table 1 <code class='latex inline'>\displaystyle \begin{array}{|c|c|} \hline Base Length, b & Perimeter, \boldsymbol{P} \\ \hline \mathbf{1} & \\ \hline 2 & \\ \hline 3 & \\ \hline 4 & \\ \hline 5 & \\ \hline \end{array} </code></p><p>Table 2 <code class='latex inline'>\displaystyle \begin{array}{|c|c|} \hline Base Length, \boldsymbol{b} & Area, \boldsymbol{A} \\ \hline 1 \\ \hline 2 \\ \hline 3 \\ \hline 4 \\ \hline 5 \\ \hline \end{array} </code></p>
<p>Find the missing terms of each arithmetic sequence. (Hint: The arithmetic mean of the first and fifth terms is the third term.)</p><p><code class='latex inline'>\displaystyle 2, a_{2}, a_{3}, a_{4},-22, \ldots </code></p>
<p>Determine the number of terms in each geometric sequence. </p><p>64, 32, 16, ..., <code class='latex inline'>\dfrac{1}{256}</code></p>
<p>Without writing the terms of the sequence. determine the general term of the geometric sequence that corresponds to each recursion formula.</p><p><code class='latex inline'>\displaystyle t_1 = \frac{5}{3}, </code></p><p><code class='latex inline'>\displaystyle t_n = (\frac{3}{4}+ c)t_{n - 1} </code></p>
<p>Given the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>, write the first three terms of the arithmetic sequence. Then, write the formula for the general term.</p><p><code class='latex inline'>a = 12, d = -5</code></p>
<p>In 1201, Leonardo Fibonacci introduced his now famous pattern of numbers called the Fibonacci sequence.</p><p><code class='latex inline'>1, 1, 2, 3, 5, 8, 13</code></p><p>Notice the pattern in this sequence. After the second number, each number in the sequence is the sum of the two numbers that precede it. That is <code class='latex inline'>2=1+1, 3=2+1, 5=3+2,</code> and so on.</p><p>Notice that every third term is divisible by 2. What do you notice about every fourth term? every fifth term?</p>
<p>Continue each pattern for three more terms. Describe how to find successive terms.</p><p><code class='latex inline'>\displaystyle 2,4,8,16 </code></p>
<p>Determine whether each sequence is geometric. If so, find the common ratio.</p><p><code class='latex inline'>\displaystyle 10,20,40, \ldots </code></p>
<p>Use the given rule to write the 4th, 5th, 6th, and 7th terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=2(n-1)^{3} </code></p>
<p>Find the missing values.</p> <ul> <li>15, 9, 3, ..., <code class='latex inline'>\bigcirc</code>, <code class='latex inline'>\bigcirc</code>, -69</li> </ul>
<p>Compare and Contrast How is finding a missing term of a geometric sequence using the geometric mean similar to finding a missing term of an arithmetic sequence using the arithmetic mean? How is it different?</p>
<p>Given the formula for the general term of an arithmetic sequence, write the first five terms. Graph the discrete function that represents each sequence.</p><p><code class='latex inline'>t_n = 0.9n + 0.3</code></p>
<p>Write the first three terms of each sequence, given the explicit formula for the nth term of the sequence.</p><p><code class='latex inline'>f(n) = 1 + 2(n - 2)</code></p>
<p>MATHEMATICAL CONNECTIONS In Exercises 47 and 48 each small square represents 1 square inch. Determine whether the areas of the figures form an arithmetic sequence. If so, write a function <code class='latex inline'>\displaystyle f </code> that represents the arithmetic sequence and find <code class='latex inline'>\displaystyle f(30) </code>.</p><p>田</p>
<p>Describe the pattern in each sequence. Write the next three terms of each sequence.</p><p>9, 7, 5, 3, ...</p>
<p>Use an arithmetic sequence to find how many multiples of 7 are between 29 and 344.</p>
<p>Determine whether each sequence is geometric. If so, find the common ratio.</p><p><code class='latex inline'>\displaystyle 18,-6,2,-\frac{2}{3}, \ldots </code></p>
<p>Find the eighth term of each sequence.</p><p><code class='latex inline'>\displaystyle 6,1,-4,-9, \ldots </code></p>
<p>Determine the first five terms of two different geometric sequences that satisfy the given conditions.</p><p>The sum of the first three terms of the sequence is 3 and the sum of the third, fourth, and fifth terms is 12.</p>
<p>Find the eighth term of each geometric sequence.</p><p><code class='latex inline'>\displaystyle 10,5,2.5, \ldots </code></p>
<p>Use appropriate tools and strategies to find the next three terms in each sequence.</p> <ul> <li>240, 120, 40, 10, 2</li> </ul>
<p>The diagrams illustrate a rule for adding odd numbers.</p><img src="/qimages/1653" /><p><strong>a)</strong> Describe the rule.</p><p><strong>b)</strong> Verify your rule for the fifth and sixth diagrams.</p><p><strong>c)</strong> Use your rule to find the sum of the odd numbers from 1 to 99.</p><p><strong>d)</strong> Use your rule to find the sum of the odd numbers from 150 to 600.</p>
<p><code class='latex inline'>\displaystyle a_{n}=n^{2}-2 n </code></p>
<p>Determine if each sequence is arithmetic, geometric, or neither. If it is arithmetic, state the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>. If it is geometric, state the values of <code class='latex inline'>a</code> and <code class='latex inline'>r</code>.</p><p><code class='latex inline'>7, 4 + s, 1 + 2s,</code> ...</p>
<p>Describe the pattern in the sequence. Give the next two terms.</p><p><code class='latex inline'> \displaystyle -6, -10, -14 </code></p>
<p>Describe the pattern in each sequence. Write the next three terms of each sequence.</p><p>300, 30, 3, 0.3, ...</p>
<p>State whether or not each sequence is arithmetic. Justify your answer.</p><p><code class='latex inline'>0.41, 0.32, 0.23</code>, ...</p>
<p>For each arithmetic sequence, determine the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>. Then, write the next four terms.</p><p>5, 8, 11, ...</p>
<p>Identify a pattern and find the next three numbers in the pattern.</p><p><code class='latex inline'>\displaystyle 1,3,9,27,81, \ldots </code></p>
<p>Identify a pattern and find the next three numbers in the pattern.</p><p><code class='latex inline'>\displaystyle 5,3,1,-1, \ldots </code></p>
<p>For each arithmetic sequence, determine the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>. Then, write the next four terms.</p><p><code class='latex inline'>\dfrac{1}{4}, \dfrac{1}{2}, \dfrac{3}{4}, ...</code></p>
<p>How many numbers between 1 and 100 are divisible by either 2 or 3 ? Explain your strategy and verify that it works.</p>
<p>Write the first four terms of each geometric sequence. </p><p><code class='latex inline'>t_n = 3(-1)^{n-1}</code></p>
<p>Write a recursion formula for each sequence.</p><p><code class='latex inline'>t_{19} = -91.8</code> and <code class='latex inline'>t_{41} = -223.8</code></p>
<p>Determine whether the sequence is arithmetic, geometric, or neither. Give a reason for your answer.</p><p>8, 8.8, 8.88, 8.888, ...</p>
<p>Think About a Plan Suppose a balloon is filled with <code class='latex inline'>\displaystyle 5000 \mathrm{~cm}^{3} </code> of helium. It then loses one fourth of its helium each day. How much helium will be left in the balloon at the start of the tenth day?</p> <ul> <li><p>How can you write a sequence of numbers to represent this situation?</p></li> <li><p>Is the sequence arithmetic, geometric, or neither?</p></li> <li><p>How can you write a formula for this sequence?</p></li> </ul>
<p>Find the unknown terms, <code class='latex inline'>m</code> and <code class='latex inline'>n</code>, in each geometric sequence.</p><p><code class='latex inline'>m, 6, n, 216</code></p>
<p>The graphs show the terms in a sequence. Write each sequence in function notation and specify the domain.</p><img src="/qimages/23294" />
<p>Write an explicit formula for each sequence. Find the tenth term.</p><p><code class='latex inline'>\displaystyle \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \ldots </code></p>
<p>For each arithmetic sequence, determine the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>. Then, write the next four terms.</p><p>0.25, 0.26, 0.27, ...</p>
<p>Determine whether the sequence is arithmetic, geometric, or neither. Give a reason for your answer.</p><p><code class='latex inline'>ab, ab^2, ab^3, ab^4, ...</code></p>
<p>Given the formula for the general term of an arithmetic sequence, write the first five terms. Graph the discrete function that represents each sequence.</p><p><code class='latex inline'>t_n = 4n - 1</code></p>
<p>The first term of a constant sequence is <code class='latex inline'>f(1) = 3.14</code>.</p><p>a) State <code class='latex inline'>f(n)</code> for this sequence.</p><p>b) Write the first five terms of this sequence.</p>
<p>An architect&#39;s starting salary is $73 000. The company has guaranteed a raise of$2275 every 6 months with satisfactory performance.</p><p>Write a recursion formula for the sequence in part a).</p>
<p>Determine whether each sequence is geometric. If so, find the common ratio.</p><p><code class='latex inline'>\displaystyle 10,15,22.5,33.75, \ldots </code></p>
<p>Determine the general term of an arithmetic sequence such that the 11th term is 53 and the sum of the 5th and 7th terms is 56.</p>
<p>Determine whether the sequence is arithmetic, geometric, or neither. Give a reason for your answer.</p><p>1, 3, 9, 27, ...</p>
<p>Write a recursive definition for each sequence.</p><p><code class='latex inline'>\displaystyle \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \ldots </code></p>
<p>Write a recursion formula for each sequence.</p><p><code class='latex inline'>t_{50} = 140</code> and <code class='latex inline'>t_{70} = 180</code></p>
<p><code class='latex inline'>\displaystyle \frac{1}{2},-\frac{1}{4}, \frac{1}{8},-\frac{1}{16}, \ldots </code></p>
<p> The first four diagrams in a pattern show arrangements made from toothpicks.</p><p>a) Copy and complete the table for the four diagrams.<code class='latex inline'>\displaystyle \begin{array}{|c|c|} \hline Diagram Number, d & Number of Toothpicks, t \\ \hline 1 \\ 2 \\ \hline 3 \\ \hline 4 \\ \hline \end{array} </code></p><p>b) Use finite differences to determine if the function is linear or quadratic.</p><p>c) Determine an equation of the function by inspection or by using your graphing calculator.</p><p>d) How many toothpicks would be needed for the arrangement shown in the 15 th diagram? the 50 th diagram?</p><img src="/qimages/156715" />
<p>Find the next two items for each pattern. Then find the 21st figure in the pattern.</p><img src="/qimages/24676" />
<p>Given the formula for the general term of an arithmetic sequence, determine <code class='latex inline'>t_{16}</code>.</p><p><code class='latex inline'>t_n = -\dfrac{3}{4}n + 3</code></p>
<p>Sketch a distance-time graph for each walker for the first 4s.</p><p>Julianne stood at a distance of 3.5 m from the sensor and did not move.</p>
<p>Identify a pattern and find the next three numbers in the pattern.</p><p><code class='latex inline'>\displaystyle 4,20,100,500, \ldots </code></p>
<p>Write a recursion formula for each sequence.</p><p><code class='latex inline'>t_{12} = 52</code> and <code class='latex inline'>t_{22} = 102</code></p>
<p>Find the first six terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=3 n^{2}-n </code></p>
<p>Describe the pattern in each sequence. Write the next three terms of each sequence.</p><p>0.11, -0.33, 0.99, - 2.97, ...</p>
<p>Write the first four terms of a sequence that satisfies each of the following. Write an explicit formula for the nth term of each of your sequences in function notation.</p><p>The terms of the sequence are determined by multiplying a constant value.</p>
<p>Find the next two terms in each sequence. Write a formula for the <code class='latex inline'>\displaystyle n </code> th term. Identify each formula as explicit or recursive.</p><p><code class='latex inline'>\displaystyle 1,8,27,64,125, \ldots </code></p>
<p>Describe the pattern in each sequence and write the next four terms.</p><p>-7, 9, -1, 2, 5, -5, 11, ...</p>
<p>Write an explicit formula for each geometric sequence. Then find the first three terms.</p><p><code class='latex inline'>\displaystyle a_{1}=1, r=2 </code></p>
<p>Describe the pattern in each sequence and write the next four terms.</p><p>2, -5, 4, -10, 6, -15, 8, -20, 10, ...</p>
<p>Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms.</p><p><code class='latex inline'>\displaystyle 1,4,9,16, \ldots </code></p>
<ol> <li>Patterns In the sequence <code class='latex inline'>\displaystyle -2,4,14,28, \ldots </code>, the number <code class='latex inline'>\displaystyle -2 </code> is in position 1 , the number 4 is in position 2 , and so on. a) Write an equation that relates each number, <code class='latex inline'>\displaystyle n </code>, to its position, <code class='latex inline'>\displaystyle p </code>, in the sequence.</li> </ol> <p>b) Repeat part a) for the sequence <code class='latex inline'>\displaystyle 2,-4,-14,-28, \ldots . </code> c) Communication How are the graphs of <code class='latex inline'>\displaystyle n </code> versus <code class='latex inline'>\displaystyle p </code> for the two sequences related?</p>
<p>Find the missing terms of each geometric sequence. (Hint: The geometric mean of the first and fifth terms is the third term. Some terms might be negative.)</p><p><code class='latex inline'>\displaystyle 2.5, \square, \square, \square, 202.5, \ldots </code></p>
<p>Write a recursion formula for each sequence.</p><p><code class='latex inline'>t_{21} = 34.5</code> and <code class='latex inline'>t_{38} = 60</code></p>
<p>Determine the number of terms in each arithmetic sequence.</p><p><code class='latex inline'>\dfrac{11}{5}, -\dfrac{4}{5}, -\dfrac{19}{5}, ..., -\dfrac{949}{5}</code></p>
<p>Continue each pattern for three more terms. Describe how to find successive terms.</p><p><code class='latex inline'>\displaystyle 4,17,30,43 </code></p>
<p>Taylor and Brooklyn are recording how far a ball rolls down a ramp during each second. The table below shows the data they have collected.</p><img src="/qimages/24670" /><p>Do the distances traveled by the ball form an arithmetic sequence? Justify your answer.</p>
<p>The graphs show the terms in a sequence. Write each sequence in function notation and specify the domain.</p><img src="/qimages/23297" />
<p>Write an explicit formula for each sequence. Then generate the first five terms.</p><p><code class='latex inline'>\displaystyle a_{1}=100, r=-20 </code></p>
<p>Write the first four terms of each geometric sequence. </p><p><code class='latex inline'>a = -2, r = \dfrac{2}{3}</code></p>
<p>Given the formula for the general term of an arithmetic sequence, determine <code class='latex inline'>t_{16}</code>.</p><p><code class='latex inline'>f(n) = 16 - 2.8n</code></p>
<p>Use appropriate tools and strategies to find the next three terms in each sequence.</p> <ul> <li><code class='latex inline'> \displaystyle \frac{3}{4}, \frac{1}{2}, \frac{1}{4} </code></li> </ul>
<p>Determine <code class='latex inline'>a</code> and <code class='latex inline'>d</code>, and then write the formula for the nth term of each arithmetic sequence with the given terms.</p><p><code class='latex inline'>t_5 = -x^3 - 6</code> and <code class='latex inline'>t_{18} = -14x^3 - 19</code></p>
<p>Given the formula for the general term of an arithmetic sequence, determine <code class='latex inline'>t_{16}</code>.</p><p><code class='latex inline'>t_n = 4n + 11</code></p>
<p>Geometry Prove that the triangle with vertices <code class='latex inline'>\displaystyle (3,5),(-2,6) </code>, and <code class='latex inline'>\displaystyle (1,3) </code> is a right triangle.</p>
<p>Write the first four terms of each geometric sequence. </p><p><code class='latex inline'>a = 22, r = -2</code></p>
<p>An architect&#39;s starting salary is $73 000. The company has guaranteed a raise of$2275 every 6 months with satisfactory performance.</p><p>Write a sequence to represent this situation. Is this sequence arithmetic? Explain.</p>
<p>Determine whether each sequence is geometric. If so, find the common ratio.</p><p><code class='latex inline'>\displaystyle 1,2,4,8, \ldots </code></p>
<p>Identify a pattern and find the next three numbers in the pattern.</p><p><code class='latex inline'>\displaystyle 1,4,3,6,5, \ldots </code></p>
<p>Determine whether the sequence is arithmetic, geometric, or neither. Give a reason for your answer.</p><p>3, 0.3, 0.03, 0.003, ...</p>
<p>Find the unknown terms, <code class='latex inline'>m</code> and <code class='latex inline'>n</code>, in each geometric sequence.</p><p><code class='latex inline'>7, m, 63, n</code></p>
<p>Marisela and Richard are finding the common difference for the arithmetic sequence -􏰀44, -􏰀32, -􏰀20, -􏰀8.</p><img src="/qimages/24668" /><p>Who is correct? Explain your reasoning.</p>
<p>Write an explicit formula for each sequence. Find the tenth term.</p><p><code class='latex inline'>\displaystyle 2,5,10,17,26, \ldots </code></p>
<p>Describe the pattern in the sequence. Give the next two terms.</p><p><code class='latex inline'> \displaystyle \frac{1}{4}, \frac{1}{2}, \frac{3}{4} </code></p>
<p>Given the formula for the general term of an arithmetic sequence, write the first five terms. Graph the discrete function that represents each sequence.</p><p><code class='latex inline'>f(n) = 2(1 - 5n)</code></p>
<p>Use each recursive definition to write an explicit formula for the sequence.</p><p><code class='latex inline'>\displaystyle a_{1}=10, a_{n}=2 a_{n-1} </code></p>
<p>Determine if each sequence is arithmetic, geometric, or neither. If it is arithmetic, state the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>. If it is geometric, state the values of <code class='latex inline'>a</code> and <code class='latex inline'>r</code>.</p><p><code class='latex inline'>3m, 7m, 11m,</code> ...</p>
<p>State the common ratio for each geometric sequence and write the next three terms.</p><p>-5, 20, -80, 320, ...</p>
<p>Chad, a champion show dog, had 2 parents one generation ago, 4 grandparents two generations ago, 8 great-grandparents three generations ago, and so on.</p><p>Write the general term of the geometric sequence that represents this situation.</p>
<p> The geometric mean of a set of 11 numbers is the nth root of the product of the numbers. </p><p>For example, given two non-consecutive terms of a geometric sequence, 6 and 24, their product is 144 and the geometric mean is <code class='latex inline'>\sqrt{144}</code>, or <code class='latex inline'>12</code>.</p><p>The numbers 6, 12, 24 form a geometric sequence. Determine the geometric mean of 8 and 128.</p>
<p>Find the next two terms in each sequence. Write a formula for the nth term. Identify each formula as explicit or recursive.</p><p><code class='latex inline'>\displaystyle 5,8,11,14,17, \ldots </code></p>
<p>For each geometric sequence, determine the formula for the general term and use it to determine the indicated term.</p><p>4096, 1024, 256, ..., <code class='latex inline'>t_8</code></p>
<p>Use each recursive definition to write an explicit formula for the sequence.</p><p><code class='latex inline'>\displaystyle a_{1}=1, a_{n}=a_{n-1}+4 </code></p>
<p>Use the given rule to write the 4th, 5th, 6th, and 7th terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=(n+1)^{2} </code></p>
<p>Find the unknown terms, <code class='latex inline'>m</code> and <code class='latex inline'>n</code>, in each geometric sequence.</p><p><code class='latex inline'>\dfrac{2}{9}, m, \dfrac{1}{2}, n</code></p>
<p>Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms.</p><p><code class='latex inline'>\displaystyle 45,90,180,360, \ldots </code></p>
<p>In an arithmetic sequence <code class='latex inline'>a_n</code>, if <code class='latex inline'>a_1=2</code> and <code class='latex inline'>a_4=11</code>, find <code class='latex inline'>a_{20}</code>.</p><p>A. 40</p><p>B. 59</p><p>C. 78</p><p>D. 87</p>
<p>Write the first three terms of each sequence, given the explicit formula for the nth term of the sequence.</p><p><code class='latex inline'>t_n = 7 + 2n</code></p>
<p>Find the eighth term of each sequence.</p><p><code class='latex inline'>\displaystyle \frac{3}{4},-\frac{3}{2}, 3,-6, \ldots </code></p>
<p>Find <code class='latex inline'>a_1</code>, for a geometric sequence with the given terms.</p><p><code class='latex inline'>\displaystyle a_{9}=\frac{1}{2} </code> and <code class='latex inline'>\displaystyle a_{12}=\frac{1}{16} </code></p>
<p>Find the next two terms in each sequence. Write a formula for the <code class='latex inline'>\displaystyle n </code> th term. Identify each formula as explicit or recursive.</p><p><code class='latex inline'>\displaystyle -1,1,-1,1,-1,1, \ldots </code></p>
<p>Each sequence has eight terms. Evaluate each related series.</p><p><code class='latex inline'>\displaystyle \frac{1}{2}, \frac{3}{2}, \frac{5}{2}, \ldots, \frac{15}{2} </code></p>
<p>Determine the number of terms in each arithmetic sequence.</p><p>-13, -9, -5, ..., 327</p>
<p>Write a recursion formula for each sequence.</p><p><code class='latex inline'>t_9 = 29 + 41 x</code> and <code class='latex inline'>t_{16} = 29 + 76x</code></p>
<p>Find the unknown terms, <code class='latex inline'>m</code> and <code class='latex inline'>n</code>, in each geometric sequence.</p><p><code class='latex inline'>-2, -10, m, n</code></p>
<p>Determine if each sequence is arithmetic, geometric, or neither. If it is arithmetic, state the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>. If it is geometric, state the values of <code class='latex inline'>a</code> and <code class='latex inline'>r</code>.</p><p><code class='latex inline'>5.440, 54.40, 544.0,</code> ...</p>
<p>Determine whether each formula is explicit or recursive. Then find the first five terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=2 n^{2}+1 </code></p>
<p>Evaluate each series to the given term.</p><p><code class='latex inline'>\displaystyle 2+4+6+8+\ldots ; </code> 10th term</p>
<p>Determine the number of terms in each arithmetic sequence.</p><p>5, 11, 17, ..., 179 </p>
<p>Determine the number of terms in each geometric sequence. </p><p>12, 4, <code class='latex inline'>\dfrac{4}{3},</code> ..., <code class='latex inline'>\dfrac{4}{729}</code></p>
<p>Write an explicit formula for each sequence. Find the tenth term.</p><p><code class='latex inline'>\displaystyle -2 \frac{1}{2},-2,-1 \frac{1}{2},-1, \ldots </code></p>
<p>Without writing the terms of the sequence, determine the general term of the geometric sequence that corresponds to each recursion formula.</p><p><code class='latex inline'>t_1 = 4, t_n = -3xt_{n-1}</code></p>
<p>Determine if each sequence is arithmetic, geometric, or neither. If it is arithmetic, state the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>. If it is geometric, state the values of <code class='latex inline'>a</code> and <code class='latex inline'>r</code>.</p><p><code class='latex inline'>\dfrac{8}{7}, \dfrac{6}{5}, \dfrac{4}{3},</code> ...</p>
<p>Find the next two items for the pattern. Then find the 16th figure in the pattern.</p><img src="/qimages/24672" />
<p>Each sequence has eight terms. Evaluate each related series.</p><p><code class='latex inline'>\displaystyle 1765,1414,1063, \ldots,-692 </code></p>
<p>Determine whether each list is a sequence or a series and finite or infinite.</p><p><code class='latex inline'>\displaystyle \frac{4}{3}, \frac{7}{3}, \frac{10}{3}, \frac{13}{3}, \frac{16}{3}, \ldots </code></p>
<p>Consider the sequence <code class='latex inline'>\sqrt 5, \sqrt {\sqrt 5}, \sqrt {\sqrt {\sqrt 5}}, ...</code></p><p>a) Write the next two terms of the sequence.</p><p>b) Express each of the five terms as a power.</p><p>c) Write an explicit formula, in function notation, to represent the terms in the sequence.</p><p>d) Express your formula in part c) in a different form.</p><p>e) Write a power to represent the 50th term in this sequence.</p>
<p>Describe the pattern in each sequence. Write the next three terms of each sequence.</p><p>5, 25, 125, 625, ...</p>
<p>Find the seventh term of each geometric sequence.</p><p><code class='latex inline'>\displaystyle 1,-3,9, \ldots </code></p>
<p>Identify a pattern and find the next three numbers in the pattern.</p><p><code class='latex inline'>\displaystyle 2,6,10,14, \ldots </code></p>
<p>Determine a formula for the nth term of each sequence.</p><p><code class='latex inline'>\dfrac{3}{8}, \dfrac{15}{24}, \dfrac{35}{48}, \dfrac{63}{80}, ...</code></p>
<p>Without writing the terms of the sequence, determine the general term of the geometric sequence that corresponds to each recursion formula.</p><p><code class='latex inline'>t_1 = -28m^3, t_n = \dfrac{1}{2}t_{n-1}</code></p>
<p>Given the formula for the general term of an arithmetic sequence, write the first five terms. Graph the discrete function that represents each sequence.</p><p><code class='latex inline'>f(n) = 0.1(3 + 2n)</code></p>
<p> The geometric mean of a set of 11 numbers is the nth root of the product of the numbers. </p><p>For example, given two non-consecutive terms of a geometric sequence, 6 and 24, their product is 144 and the geometric mean is <code class='latex inline'>\sqrt{144}</code>, or <code class='latex inline'>12</code>.</p><p>The numbers 6, 12, 24 form a geometric sequence. Determine the geometric mean of 2 and 200.</p>
<p>For each sequence, make a table of values using the term number and term, and calculate the finite differences. Then, determine an explicit formula in function notation and specify the domain.</p><p>4, 13, 26, 43, ...</p>
<p>Given the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>, write the first three terms of the arithmetic sequence. Then, write the formula for the general term.</p><p><code class='latex inline'>a = -9, d = 2</code></p>
<p>The cross section of a honeycomb can be thought of as a central regular hexagon surrounded by a ring of six more hexagons, surrounded by a ring of twelve more hexagons, and so on. Think of the central hexagon as ring 1 , the ring of six hexagons as ring 2 , and so on.</p><p>a) Copy and complete the table by recording the total numbers of hexagons in the first ring, the first two rings, the first three rings, and the first four rings.</p><img src="/qimages/156716" /><p>b) Use finite differences to determine whether the function is linear or quadratic.</p><p>c) Determine an equation of the function by using your graphing calculator. d) Find the total number of hexagons in the first ten rings.</p>
<p>Identify the pattern and find the next three terms.</p><p><code class='latex inline'>\displaystyle 10,8,6,4,2,0, \ldots </code></p>
<p>For each geometric sequence, determine the formula for the general term and use it to determine the indicated term.</p><p>6, -2, <code class='latex inline'>\dfrac{2}{3}</code>, ..., <code class='latex inline'>t_8</code></p>
<p>State whether or not each sequence is arithmetic. Justify your answer.</p><p><code class='latex inline'>\dfrac{19}{12}, \dfrac{5}{4}, \dfrac{11}{12}, ...</code></p>
<p>Given the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>, write the first three terms of the arithmetic sequence. Then, write the formula for the general term.</p><p><code class='latex inline'>a = 11, d = -\dfrac{7}{8}</code></p>
<p>Given the formula for the general term of an arithmetic sequence, write the first five terms. Graph the discrete function that represents each sequence.</p><p><code class='latex inline'>f(n) = -n + 3</code></p>
<p>Find the next two terms in each sequence. Write a formula for the <code class='latex inline'>\displaystyle n </code> th term. Identify each formula as explicit or recursive.</p><p><code class='latex inline'>\displaystyle 49,64,81,100,121, \ldots </code></p>
<p>Write the first three terms of each sequence, given the explicit formula for the nth term of the sequence.</p><p><code class='latex inline'>f(n) = -2(3)^{n + 1}</code></p>
<p>Describe the pattern in each sequence. Write the next three terms of each sequence.</p><p>-4, -8, -16, -32, ...</p>
<p>Write the first four terms of each geometric sequence. </p><p><code class='latex inline'>f(n) = 4(\sqrt{5})^{n-1}</code></p>
<p>Write an explicit formula for each sequence. Then generate the first five terms.</p><p><code class='latex inline'>\displaystyle a_{1}=1, r=0.5 </code></p>
<p>Describe the pattern in the sequence. Give the next two terms.</p><p><code class='latex inline'> \displaystyle 15, 10, 5 </code></p>
<p>Suppose you arrange a number of regular pentagons so that only one side of each pentagon touches. Each side of each pentagon is 1 centimeter.</p><img src="/qimages/24685" /><p>For each arrangement of pentagons, compute the perimeter.</p>
<p>To find the third term of the geometric sequence <code class='latex inline'>\displaystyle 5,10, \square,\square, 80 </code>, your friend says that there are two possible answers - the geometric mean of 5 and 80, and its opposite. Explain your friend&#39;s error.</p>
<p>Write an explicit formula for each sequence. Then generate the first five terms.</p><p><code class='latex inline'>\displaystyle a_{1}=10, r=-1 </code></p>
<p>Write the 16th term, given the explicit formula for the nth term of the sequence.</p><p><code class='latex inline'>t_n = 3n + 2</code></p>
<p>Write the first three terms of each sequence, given the explicit formula for the nth term of the sequence.</p><p><code class='latex inline'>t_n = 2n + 1</code></p>
<p>Find the first six terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=2^{n}-1 </code></p>
<p>Write a recursive definition for each sequence.</p><p><code class='latex inline'>\displaystyle 4,-8,16,-32,64, \ldots </code></p>
<ol> <li>Square pattern The first four diagrams in a pattern are made up of small green squares and white squares of side length 1 unit.</li> </ol> <p>a) Copy and complete the table for the four diagrams.</p><img src="/qimages/156711" /><p>Side Length of Large Square, <code class='latex inline'>\displaystyle s </code> Number of Small White Squares, <code class='latex inline'>\displaystyle n </code></p><p><code class='latex inline'>\displaystyle \begin{array}{l} 2 \\ 3 \\ 4 \\ 5 \end{array} </code></p><p>b) Use finite differences to determine if the function is linear or quadratic.</p><p>c) Determine an equation of the function by inspection or by using your graphing calculator.</p><p>d) How many small white squares will there be in a large square of side length 10 units? 25 units?</p><img src="/qimages/156708" />
<p>Write an explicit formula for each sequence. Find the tenth term.</p><p><code class='latex inline'>\displaystyle 3,7,11,15,19, \ldots </code></p>
<p>On an island cruise in Hawaii, each passenger is given a flower chain. A crew member hands out 3 red, 3 blue, and 3 green chains in that order. If this pattern is repeated, what color chain will the 50th person receive?</p>
<p>Describe the pattern in the sequence. Give the next two terms.</p><img src="/qimages/152699" />
<p>A new small business plans to triple its sales every day for the first 2 weeks. Sales on the first day are $20.</p><p>a) Write the sequence that represents the sales for the first 6 days according to the plan.</p><p>b) Write an explicit formula to determine the sales on any of the first 14 days.</p><p>c) Use your formula to determine the sales on the 14th day. Is this reasonable? Why or why not?</p> <p>Find the missing term of each geometric sequence. It could be the geometric mean or its opposite.</p><p><code class='latex inline'>\displaystyle 3, \square, 0.75, \ldots </code></p> <p>State the common ratio for each geometric sequence and write the next three terms.</p><p>3, 6, 12, 24, ...</p> <p>Find the eighth term of each sequence.</p><p><code class='latex inline'>\displaystyle \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \ldots </code></p> <p>The product of the first two terms of a geometric sequence is 27, and the product of the first three terms is also 27.</p><p> State the general term of the sequence.</p> <p>State whether or not each sequence is arithmetic. Justify your answer.</p><p><code class='latex inline'>-\dfrac{5}{4}, -\dfrac{4}{3}, -\dfrac{3}{2}, ...</code></p> <p>Determine an explicit formula for the nth term of each sequence. Use the formula to write the 15th term.</p><p><code class='latex inline'>1, \dfrac{2}{3}, \dfrac{3}{5}, \dfrac{4}{7}, ...</code></p> <p>Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms.</p><p><code class='latex inline'>\displaystyle 2,1,0.5,0.25, \ldots </code></p> <p>Use the given rule to write the 4th, 5th, 6th, and 7th terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=\frac{n^{2}}{n+1} </code></p> <p>Determine a formula for the nth term of each sequence.</p><p><code class='latex inline'>1 \times 1, 3 \times 4, 5 \times 7, 7 \times 10, ...</code></p> <p>Determine whether each formula is explicit or recursive. Then find the first five terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=(n-5)(n+5) </code></p> <p>Find the next two terms in each sequence. Write a formula for the nth term. Identify each formula as explicit or recursive.</p><p><code class='latex inline'>\displaystyle 3,6,12,24,48, \ldots </code></p> <p>Write a recursive definition for each sequence.</p><p><code class='latex inline'>\displaystyle 0,3,7,12,18, \ldots </code></p> <p>Find the eighth term of each geometric sequence.</p><p><code class='latex inline'>\displaystyle -3,6,-12, \ldots </code></p> <p>Find the nth term of each arithmetic sequence described.</p><p><code class='latex inline'>-7, -3, 1, 5,...</code> for <code class='latex inline'>n=35</code></p> <p>For each geometric sequence, determine the formula for the general term and use it to determine the indicated term.</p><p><code class='latex inline'>\dfrac{a^2}{b}, \dfrac{a^3}{2b}, \dfrac{a^4}{4b}</code>, ..., <code class='latex inline'>t_16</code></p> <p>Determine whether the sequence is arithmetic, geometric, or neither. Give a reason for your answer.</p><p>4, -16, 64, -256, ...</p> <p>Find the missing terms of each geometric sequence. (Hint: The geometric mean of the first and fifth terms is the third term. Some terms might be negative.)</p><p><code class='latex inline'>\displaystyle 972, \square, \square, \square, 12, \ldots </code></p> <p>Find the next two terms in each sequence. Write a formula for the <code class='latex inline'>\displaystyle n </code> th term. Identify each formula as explicit or recursive.</p><p><code class='latex inline'>\displaystyle -75,-68,-61,-54, \ldots </code></p> <p>Identify a pattern and find the next three numbers in the pattern.</p><p><code class='latex inline'>\displaystyle 3,6,10,15, \ldots </code></p> <p>An original painting is purchased for$230, and each year it increases in value by 22% of its original value.</p><p>When is the painting worth $1242?</p> <p>For each sequence, make a table of values using the term number and term, and calculate the finite differences. Then, determine an explicit formula in function notation and specify the domain.</p><p>7, 4, 1, -2, ...</p> <p>Find the unknown terms, <code class='latex inline'>m</code> and <code class='latex inline'>n</code>, in each geometric sequence.</p><p><code class='latex inline'>4, m, n, 500</code></p> <p>Find the first six terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=-5 n+1 </code></p> <p>Use appropriate tools and strategies to find the next three terms in each sequence.</p> <ul> <li><code class='latex inline'> \displaystyle 0, -\frac{1}{3}, -\frac{2}{3}, -1 </code></li> </ul> <p>Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms in the sequence.</p><p><code class='latex inline'>8, 9, 10, 11,...</code></p> <p>State the common ratio for each geometric sequence and write the next three terms.</p><p><code class='latex inline'>-\dfrac{1}{6}, -\dfrac{1}{2}, -\dfrac{3}{2}, -\dfrac{9}{2}, ...</code></p> <p>Determine the first three terms of a geometric sequence such that the sum of the second and third terms is 24 and the sum of the seventh and eighth terms is 5832.</p> <p>Describe the pattern in each sequence. Write the next three terms of each sequence.</p><p>3, 9, 27, 81, 243, ...</p> <p>Write an explicit formula for each sequence. Then generate the first five terms.</p><p><code class='latex inline'>\displaystyle a_{1}=1024, r=0.5 </code></p> <p>Find the next three terms of each arithmetic sequence.</p><p><code class='latex inline'>-25, -19, -13, -7,...</code></p> <p>Each sequence has eight terms. Evaluate each related series.</p><p><code class='latex inline'>\displaystyle 5,13,21, \ldots, 61 </code></p> <p>Find the missing values.</p> <ul> <li>-5, -8, -11, ..., <code class='latex inline'>\bigcirc</code>, <code class='latex inline'>\bigcirc</code>, -164</li> </ul> <p>Find the first six terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=\frac{1}{2} n </code></p> <p>Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms in the sequence.</p><p><code class='latex inline'>2, 8, 14, 20,...</code></p> <p>Which term of the geometric sequence <code class='latex inline'>\dfrac{2}{81}, \dfrac{4}{27}, \dfrac{8}{9}</code>, ... has a value of 6912?</p> <p>Find the first six terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=n^{2}+1 </code></p> <p>Identify a pattern and find the next three numbers in the pattern.</p><p><code class='latex inline'>\displaystyle 30,45,60,75, \ldots </code></p> <p>Find the missing term of each geometric sequence. It could be the geometric mean or its opposite.</p><p><code class='latex inline'>\displaystyle 5, \square, 911.25, \ldots </code></p> <p>Determine if each sequence is arithmetic, geometric, or neither. If it is arithmetic, state the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>. If it is geometric, state the values of <code class='latex inline'>a</code> and <code class='latex inline'>r</code>.</p><p><code class='latex inline'>3x, -4y, 5x - 6y, 7x - 8y</code> ...</p> <p>In a bacteria strain, the number of bacteria doubles every 20 min. There were 8 bacteria to start with.</p><p> Write an expression to represent the term that corresponds to the number of bacteria after 1 day.</p> <p>Find the next three terms of each arithmetic sequence.</p><p><code class='latex inline'>1, 4, 7, 10,...</code></p> <p>Write the first four terms of each geometric sequence. </p><p><code class='latex inline'>f(n) = \dfrac{1}{3}(2)^{n-1}</code></p> <p>Determine whether each sequence is geometric. If so, find the common ratio.</p><p><code class='latex inline'>\displaystyle \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \ldots </code></p> <p>Determine an explicit formula for the nth term of each sequence. Use the formula to write the 15th term.</p><p>1, 3, 9, 27, ...</p> <p>Find the missing values.</p> <ul> <li>-1024, 512, -2256, ..., <code class='latex inline'>\bigcirc</code>, <code class='latex inline'>\bigcirc</code>, -1</li> </ul> <p>Use the given rule to write the 4th, 5th, 6th, and 7th terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=\frac{n+1}{n+2} </code></p> <p>Find the seventh term of each geometric sequence.</p><p><code class='latex inline'>\displaystyle 100,20,4, \ldots </code></p> <p>Write a recursive definition for each sequence.</p><p><code class='latex inline'>\displaystyle 1,2,6,24,120, \ldots </code></p> <p>Write the 16th term, given the explicit formula for the nth term of the sequence.</p><p><code class='latex inline'>f(n) = (-2)^{10 - n}</code></p> <p>Which term in the arithmetic sequence -15, -8, -1, ... has the value 125?</p> <p>Describe the pattern in the sequence. Give the next two terms.</p><p><code class='latex inline'>\displaystyle -96,-48,-24 </code></p> <p>Determine the number of terms in each arithmetic sequence.</p><p>8, 3, -2, ..., -152</p> <p>Continue each pattern for three more terms. Describe how to find successive terms.</p> <ul> <li>1, 1, 2, 3, 5, 8</li> </ul> <p>MATHEMATICAL CONNECTIONS In Exercises 47 and 48 each small square represents 1 square inch. Determine whether the areas of the figures form an arithmetic sequence. If so, write a function <code class='latex inline'>\displaystyle f </code> that represents the arithmetic sequence and find <code class='latex inline'>\displaystyle f(30) </code>.</p><img src="/qimages/44667" /> <p>For each sequence, make a table of values using the term number and term, and calculate the finite differences. Then, determine an explicit formula in function notation and specify the domain.</p><p>6, 12, 18, 24, ...</p> <p>Write the first three terms of each sequence, given the explicit formula for the nth term of the sequence.</p><p><code class='latex inline'>t_n = 2^{n - 1}</code></p> <p>Describe the pattern in each sequence and write the next four terms.</p><p><code class='latex inline'>4, \dfrac{1}{27}, 9, \dfrac{1}{9}, 14, \dfrac{1}{3}, 19, ...</code></p> <p>Find the missing values.</p> <ul> <li>3, -6, 12, ..., <code class='latex inline'>\bigcirc</code>, <code class='latex inline'>\bigcirc</code>, -24 576</li> </ul> <p>Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms.</p><p><code class='latex inline'>\displaystyle -5,10,-20,40, \ldots </code></p> <p>Given the formula for the general term of an arithmetic sequence, write the first five terms. Graph the discrete function that represents each sequence.</p><p><code class='latex inline'>t_n = -6n + 17</code></p> <p>Write a recursion formula for each sequence.</p><p><code class='latex inline'>t_{34} = 96</code> and <code class='latex inline'>t_{46} = 132</code></p> <p>Describe the pattern in the sequence. Give the next two terms.</p><p><code class='latex inline'> \displaystyle 100, 80, 65, 55 </code></p> <p>Identify each sequence as arithmetic, geometric, or neither. Then find the next two terms.</p><p><code class='latex inline'>\displaystyle 25,50,75,100, \ldots </code></p> <p>Find the missing values.</p> <ul> <li>-400, -376, -352, ..., <code class='latex inline'>\bigcirc</code>, <code class='latex inline'>\bigcirc</code>, 80</li> </ul> <p>An architect&#39;s starting salary is$73 000. The company has guaranteed a raise of $2275 every 6 months with satisfactory performance.</p><p>State the general term of the sequence in part a).</p> <p><code class='latex inline'>\displaystyle a_{n}=5 n-3 </code></p> <p>Determine if each sequence is arithmetic, geometric, or neither. If it is arithmetic, state the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>. If it is geometric, state the values of <code class='latex inline'>a</code> and <code class='latex inline'>r</code>.</p><p><code class='latex inline'>-1, -\dfrac{2}{x}, -\dfrac{4}{x^2},</code> ...</p> <p>A student claims that the next term of the arithmetic sequence <code class='latex inline'>\displaystyle 0,2,4, \ldots </code> is 8 . Explain and correct the student&#39;s error.</p> <p>Determine the number of terms in each geometric sequence. </p><p>5, 35, 245, ..., 588 245</p> <p>Find the eighth term of each sequence.</p><p><code class='latex inline'>\displaystyle -2,-1,0,1,2, \ldots </code></p> <p>Find the eighth term of each sequence.</p><p><code class='latex inline'>\displaystyle 2,1,-2,-7,-14, \ldots </code></p> <p>Use an appropriate tool to help determine the thousandth term in the sequence 45, 41, 37, 33, ... .</p> <p>A number, <code class='latex inline'>m</code>, is called an arithmetic mean between <code class='latex inline'>a</code> and <code class='latex inline'>b</code> if <code class='latex inline'>a</code>, <code class='latex inline'>m</code>, and <code class='latex inline'>b</code> form an arithmetic sequence.</p><p>Determine the arithmetic mean between 3 and 27.</p> <p>Determine the value of <code class='latex inline'>y</code> if <code class='latex inline'>y - 2</code>, <code class='latex inline'>5y + 10</code>, and <code class='latex inline'>y - 50</code> are consecutive terms in a geometric sequence.</p> <p>Write a recursive definition for each sequence.</p><p><code class='latex inline'>\displaystyle 80,77,74,71,68, \ldots </code></p> <p>Find the value of <code class='latex inline'>y</code> that makes <code class='latex inline'>y+4, 6, y, ...</code> an arithmetic sequence.</p> <p>Find the eighth term of each sequence.</p><p><code class='latex inline'>\displaystyle 43,41,39,37,35, \ldots </code></p> <p>Write the first four terms of each geometric sequence. </p><p><code class='latex inline'>t_n = -1111(0.3)^{n-1}</code></p> <p>State the common ratio for each geometric sequence and write the next three terms.</p><p>2.5, 0.5, 0.1, 0.02, ...</p> <p>Find the next two terms in each sequence. Write a formula for the <code class='latex inline'>\displaystyle n </code> th term. Identify each formula as explicit or recursive.</p><p><code class='latex inline'>\displaystyle 4,16,64,256,1024, \ldots </code></p> <p>Find the eighth term of each geometric sequence.</p><p><code class='latex inline'>\displaystyle 3,9,27, \ldots </code></p> <p>Find the next three terms of each arithmetic sequence.</p><p><code class='latex inline'>9, 5, 1, -3,...</code></p> <p>Write an equation for the nth term of each arithmetic sequence. Then graph the first five terms in the sequence.</p><p><code class='latex inline'>-3, -6, -9, -12,...</code></p> <p>Find the next two terms in the sequence <code class='latex inline'>3, 4, 6, 9,...</code></p><p>A. 12, 15</p><p>B. 13, 18</p><p>C. 14, 19</p><p>D. 15, 21</p> <p>Determine whether each formula is explicit or recursive. Then find the first five terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=-4 n^{2}-2 </code></p> <p>Identify a pattern and draw the next three figures in the pattern.</p><img src="/qimages/87289" /> <p>Continue each pattern for three more terms. Describe how to find successive terms.</p><p><code class='latex inline'>\displaystyle \begin{array}{ll}\text { a) } 12,9,6,3 & \text { b) } 7,14,28 \\ \text { c) } 5,6,8,11 & \text { d) } 3,-1,-6,-12\end{array} </code></p> <p>Find the first six terms of each sequence.</p><p><code class='latex inline'>\displaystyle a_{n}=3 n+2 </code></p> <p>Use each circle to find the length of the indicated arc. Round your answer to the nearest tenth.</p><p>(4)</p> <p>Find the next three terms of each arithmetic sequence.</p><p><code class='latex inline'>22, 34, 46, 58,...</code></p> <p>Write an explicit formula for each sequence. Find the tenth term.</p><p><code class='latex inline'>\displaystyle 1,3,9,27, \ldots </code></p> <p>Given the formula for the general term of an arithmetic sequence, determine <code class='latex inline'>t_{16}</code>.</p><p><code class='latex inline'>t_n = 2n + 7</code></p> <p>Describe the pattern in the sequence. Give the next two terms.</p><p><code class='latex inline'> \displaystyle 3, -6, 12, -24 </code></p> <p>Use each recursive definition to write an explicit formula for the sequence.</p><p><code class='latex inline'>\displaystyle a_{1}=-5, a_{n}=a_{n-1}-1 </code></p> <p>Evaluate each series to the given term.</p><p><code class='latex inline'>\displaystyle -5-25-45-\ldots ; 9 </code> th term</p> <p>Find the next two terms in each sequence. Write a formula for the <code class='latex inline'>\displaystyle n </code> th term. Identify each formula as explicit or recursive.</p><p><code class='latex inline'>\displaystyle -16,-8,-4,-2, \ldots </code></p> <p>Write a recursive definition for each sequence.</p><p><code class='latex inline'>\displaystyle 4,8,16,32,64, \ldots </code></p> <p>For each arithmetic sequence, determine the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>. Then, write the next four terms.</p><p>33, 31.2, 29.4, ...</p> <p>Write an explicit formula for each sequence. Find the tenth term.</p><p><code class='latex inline'>\displaystyle 4,5,6,7,8, \ldots </code></p> <p>Write the 16th term, given the explicit formula for the nth term of the sequence.</p><p>a) <code class='latex inline'>f(n) = n^2 - 6</code></p><p>b) <code class='latex inline'>t_n = \dfrac{n-2}{n}</code></p> <p>Find the eighth term of each sequence.</p><p><code class='latex inline'>\displaystyle 144,36,9, \frac{9}{4}, \ldots </code></p> <p>Find the missing terms of each geometric sequence. (Hint: The geometric mean of the first and fifth terms is the third term. Some terms might be negative.)</p><p><code class='latex inline'>\displaystyle -4, \square, \square, \square,-30 \frac{3}{8}, \ldots </code></p> <p>Describe the pattern in each sequence and write the next four terms.</p><p>8, -1, 0.8, -1, 0.08, -1, ...</p> <p>Identify the pattern and find the next three terms.</p><p><code class='latex inline'>\displaystyle 100,117,134,151,168, \ldots </code></p> <p>Consider the sequence 9, 18, 27, 36, ... Determine whether each of the following numbers is part of this sequence. Explain your thinking.</p><p>a) 135</p><p>b) 179</p><p>c) 653</p><p>d) 423</p> <p>Determine the value of <code class='latex inline'>y</code> if <code class='latex inline'>y - 2</code>, <code class='latex inline'>5y + 10</code>, and <code class='latex inline'>y - 50</code> are consecutive terms in a arithemetic sequence.</p> <p>Determine the first three terms of a geometric sequence such that the sum of the second and tthd terms is 24 and the sum of the seventh and eighth terms is 5832.</p> <p>Find the nth term of each arithmetic sequence described.</p><p><code class='latex inline'>200</code> is the <code class='latex inline'>\underline{?}</code>th term of <code class='latex inline'>24, 35, 46, 57,...</code></p> <p>Find the missing term of each geometric sequence. It could be the geometric mean or its opposite.</p><p><code class='latex inline'>9180, \square , 255, \ldots </code></p> <p>Find the eighth term of each sequence.</p><p><code class='latex inline'>\displaystyle 2,-\frac{3}{2}, \frac{4}{3},-\frac{5}{4}, \ldots </code></p> <p>State whether or not each sequence is arithmetic. Justify your answer.</p><p><code class='latex inline'>3, 2, 3, 4</code>, ...</p> <p>Write the first four terms of a sequence that satisfies each of the following. Write an explicit formula for the nth term of each of your sequences in function notation.</p><p>The terms of the sequence are determined by subtracting a constant value.</p> <p>Identify a pattern and find the next three numbers in the pattern.</p><p><code class='latex inline'>\displaystyle 144,132,120,108, \ldots </code></p> <p>Find the eighth term of each sequence.</p><p><code class='latex inline'>\displaystyle 40,20,10,5, \frac{5}{2}, \ldots </code></p> <p>For each geometric sequence, determine the formula for the general term and use it to determine the indicated term.</p><p>12, 6, 3, ..., <code class='latex inline'>t_12</code></p> <p>Find the eighth term of each geometric sequence.</p><p><code class='latex inline'>\displaystyle 24,-6, \frac{3}{2}, \ldots </code></p> <p>Write the first three terms of each sequence, given the explicit formula for the nth term of the sequence.</p><p><code class='latex inline'>t_n = 4 - 3n</code></p> <p>Write a recursion formula for each sequence.</p><p><code class='latex inline'>t_3 = 11</code> and <code class='latex inline'>t_8 = 46</code></p> <p>Find the next two terms in each sequence. Write a formula for the <code class='latex inline'>\displaystyle n </code> th term. Identify each formula as explicit or recursive.</p><p><code class='latex inline'>\displaystyle 21,13,5,-3, \ldots </code></p> <p>Write a recursive definition for each sequence.</p><p><code class='latex inline'>\displaystyle 1,4,7,10,13, \ldots </code></p> <p>Given the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>, write the first three terms of the arithmetic sequence. Then, write the formula for the general term.</p><p><code class='latex inline'>a = -\dfrac{2}{3}, d = \dfrac{1}{2}</code></p> <p>The graphs show the terms in a sequence. Write each sequence in function notation and specify the domain.</p><img src="/qimages/23295" /> <p>Find the missing term of each geometric sequence. It could be the geometric mean or its opposite.</p><p><code class='latex inline'>\displaystyle 12, \square, 3, \ldots </code></p> <p>Given the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>, write the first three terms of the arithmetic sequence. Then, write the formula for the general term.</p><p><code class='latex inline'>a = x^2, d = 1.3x^2</code></p> <p>Determine the number of terms in each arithmetic sequence.</p><p>8, 16, 24, ..., 424</p> <p>Determine whether each sequence is geometric. If so, find the common ratio.</p><p><code class='latex inline'>\displaystyle 1,-2,4,-8, \ldots </code></p> <p>Determine whether each sequence is geometric. If so, find the common ratio.</p><p><code class='latex inline'>\displaystyle 1,2,3,4, \ldots </code></p> <p>Write an explicit formula for each sequence. Find the tenth term.</p><p><code class='latex inline'>\displaystyle 1,4,9,16, \ldots </code></p> <p>For each arithmetic sequence, determine the values of <code class='latex inline'>a</code> and <code class='latex inline'>d</code>. Then, write the next four terms.</p><p>1.5, 0.7, -0.1, ...</p> <p>Determine all possible arithmetic sequences formed by the numbers <code class='latex inline'>p, q</code>, and <code class='latex inline'>r</code> such that <code class='latex inline'>q = 3</code> and <code class='latex inline'>p^2 + r^2 =68</code></p> <p>Determine whether each sequence is geometric. If so, find the common ratio.</p><p><code class='latex inline'>\displaystyle 5,10,15, \ldots </code></p> <p>State the common ratio for each geometric sequence and write the next three terms.</p><p><code class='latex inline'>\dfrac{(x + 3)^2}{3}, \dfrac{(x + 3)^5}{12}, \dfrac{(x + 3)^8}{48}, \dfrac{(x + 3)^{11}}{192}, ...</code></p> <p>For each geometric sequence, determine the formula for the general term and use it to determine the indicated term.</p><p><code class='latex inline'>\dfrac{1}{32}, \dfrac{1}{8}, \dfrac{1}{2}</code>, ..., <code class='latex inline'>t_13</code></p> <p>Find the 17 th term of the arithmetic sequence.</p><p><code class='latex inline'>\displaystyle a_{18}=-9, d=-11 </code></p> <p>Write a recursive definition for each sequence.</p><p><code class='latex inline'>\displaystyle 1,5,14,30, \ldots </code></p> <p>An architect’s starting salary is$73 000. The company has guaranteed a raise of $2275 every 6 months with satisfactory performance.</p><p>When will the architect’s salary be$127 600?</p>
<p>Find the missing values.</p> <ul> <li>5, 15, 45, ..., <code class='latex inline'>\bigcirc</code>, <code class='latex inline'>\bigcirc</code>, 10 935</li> </ul>
<p>Chad, a champion show dog, had 2 parents one generation ago, 4 grandparents two generations ago, 8 great-grandparents three generations ago, and so on.</p><p>Determine how many ancestors Chad had each number of generations ago.</p><p>i) 6 ii) 10 iii) 14</p>
<p>Identify the pattern and find the next three terms.</p><p><code class='latex inline'>\displaystyle \frac{5}{7}, \frac{8}{7}, \frac{11}{7}, 2, \ldots </code></p>
<p>In 1201, Leonardo Fibonacci introduced his now famous pattern of numbers called the Fibonacci sequence.</p><p><code class='latex inline'>1, 1, 2, 3, 5, 8, 13</code></p><p>Notice the pattern in this sequence. After the second number, each number in the sequence is the sum of the two numbers that precede it. That is <code class='latex inline'>2=1+1, 3=2+1, 5=3+2,</code> and so on.</p><p>Write the first 12 terms of the Fibonacci sequence.</p>
<p>Write the 16th term, given the explicit formula for the nth term of the sequence.</p><p><code class='latex inline'>t_n = \sqrt n + 2</code></p>
<p>Given the formula for the general term of an arithmetic sequence, determine <code class='latex inline'>t_{16}</code>.</p><p><code class='latex inline'>f(n) = 3 - 5n</code></p>
<p>An original painting is purchased for \$230, and each year it increases in value by 22% of its original value.</p><p>What is the painting&#39;s value after 12 years?</p>
<p>Identify a pattern and find the next three numbers in the pattern.</p><p><code class='latex inline'>\displaystyle 8,16,24,32, \ldots </code></p>
<p>Find the missing terms of each arithmetic sequence. (Hint: The arithmetic mean of the first and fifth terms is the third term.)</p><p><code class='latex inline'>\displaystyle 17, a_{2}, a_{3}, a_{4}, 17, \ldots </code></p>
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