7.6 Geometric Series
Chapter
Chapter 7
Section
7.6
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Solutions 39 Videos

Calculate the sum of the first seven terms of each geometric series.

6+18+54+...

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0.40mins
Q1a

Calculate the sum of the first seven terms of each geometric series.

100+50+25+...

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0.39mins
Q1b

Calculate the sum of the first seven terms of each geometric series.

8-24+72-...

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0.38mins
Q1c

Calculate the sum of the first seven terms of each geometric series.

\displaystyle{\frac{1}{3}+\frac{1}{6}+\frac{1}{12}+...}

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0.46mins
Q1d

Calculate the sum of the first six terms of a geometric sequence with first term 11 and common ratio 4.

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0.42mins
Q2

For each geometric series, calculate t_6 and S_6

6+30+150+...

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0.46mins
Q3a

For each geometric series, calculate t_6 and S_6

-11-33-99-...

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0.43mins
Q3b

For each geometric series, calculate t_6 and S_6

21 000 000+4 200 000+840 000+...

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0.46mins
Q3c

For each geometric series, calculate t_6 and S_6

\displaystyle{\frac{4}{5}+\frac{8}{15}+\frac{16}{45}+...}

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1.30mins
Q3d

For each geometric series, calculate t_6 and S_6

3.4-7.14+14.994-...

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1.13mins
Q3e

For each geometric series, calculate t_6 and S_6

1+3x^2+9x^4

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2.05mins
Q3f

i) Determine whether each series is arithmetic, geometric, or neither.

ii) If the series is geometric, calculate the sum of the first eight terms.

5+10+15+20+...

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0.13mins
Q4a

i) Determine whether each series is arithmetic, geometric, or neither.

ii) If the series is geometric, calculate the sum of the first eight terms.

7+21+63+189+...

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0.35mins
Q4b

i) Determine whether each series is arithmetic, geometric, or neither.

ii) If the series is geometric, calculate the sum of the first eight terms.

2048-512+128-32+...

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1.23mins
Q4c

i) Determine whether each series is arithmetic, geometric, or neither.

ii) If the series is geometric, calculate the sum of the first eight terms.

10-20+30-40+...

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0.11mins
Q4d

i) Determine whether each series is arithmetic, geometric, or neither.

ii) If the series is geometric, calculate the sum of the first eight terms.

1.1+1.21+1.331+1.4641+...

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0.32mins
Q4e

i) Determine whether each series is arithmetic, geometric, or neither.

ii) If the series is geometric, calculate the sum of the first eight terms.

81+63+45+27+...

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0.38mins
Q4f

Determine the sum of the first seven terms of the geometric series in which

t_1=13 and r=5

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0.19mins
Q5a

Determine the sum of the first seven terms of the geometric series in which the first term is 11 and the seventh term is 704

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1.37mins
Q5b

Determine the sum of the first seven terms of the geometric series in which t_1=120 and t_2=30

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1.14mins
Q5c

Determine the sum of the first seven terms of the geometric series in which the third term is 18 and the terms increase by a factor of 3

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0.38mins
Q5d

Determine the sum of the first seven terms of the geometric series in which t_8=1024 and the terms decrease by a factor of \displaystyle{\frac{2}{3}}

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1.26mins
Q5e

Determine the sum of the first seven terms of the geometric series in which t_5=5 and t_8=-40

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1.29mins
Q5f

Calculate the sum of each geometric series.

1+6+36+...+279 936

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1.00mins
Q6a

Calculate the sum of each geometric series.

960+480+240+...+15

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2.02mins
Q6b

Calculate the sum of each geometric series.

17-51+153-...-334 611

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1.30mins
Q6c

Calculate the sum of each geometric series.

24 000+3600+540+...+1.8225

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2.06mins
Q6d

Calculate the sum of each geometric series.

-6+24-96+...+98 304

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1.33mins
Q6e

Calculate the sum of each geometric series.

\displaystyle{4+2+1+...+\frac{1}{1024}}

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2.22mins
Q6f

A ball is dropped from a height of 3 m and bounces on the ground. At the top of each bounce, the ball reaches 60% of its previous height. Calculate the total distance travelled by the ball when it hits the ground for the fifth time.

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2.34mins
Q7

The formula for the sum of a geometric series is \displaystyle{S_n=\frac{a(r^n-1)}{r-1}} or \displaystyle{S_n=\frac{t_{n+1}-t_1}{r-1}}, each of which is valid only if r\neq1. Explain how you would determine the sum of a geometric series if r=1

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1.18mins
Q8

A Pythagorean fractal tree starts at stage 1 with a square of side length 1 m. At every consecutive stage, an isosceles right triangle and two squares are attached to the last square(s) drawn. The first three stages are shown. Calculate the area of the tree at the 10th stage.

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4.42mins
Q10

A large company has a phone tree to contact its employees in case of an emergency factory shutdown. Each of the five senior managers calls three employees, who each call three other employees, and so on. If the tree consists of seven levels, how many employees does the company have?

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1.14mins
Q11

John wants to calculate the sum of a geometric series with 10 terms, where the 10th term is 5\ 859\ 375 and the commom ratio is \dfrac{5}{3}. John solved the problem by considering another geometric series with common ratio \dfrac{3}{5}. Use John’s method to calculate the sum. Justify your reasoning.

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3.22mins
Q12

A cereal company attempts to promote its product by placing certificates for a cash prize in selected boxes. The company wants to come up with a number of prizes that satisfy all of these conditions:

a) The total of the prizes is at most $2\ 000\ 000.

b) Each prize is in whole dollars (no cents).

c) When the prizes are arranged from least to greatest, each prize is a constant integral multiple of the next smaller prize and is

  • more than double the next smaller prize

  • less than 10 times the next smaller prize Determine a set of prizes that satisfies these conditions.

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3.46mins
Q13

In a geometric series, t_1=12 and S_3=372. What is the greatest possible value for t_5? Justify your answer.

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2.04mins
Q15

In a geometric series, t_1=23, t_3=92, and the sum of all of the terms of the series is 62 813. How many terms are in the series?

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2.13mins
Q16

Factor x^{15} - 1.

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1.31mins
Q17
Lecture on Geometric Series 8 Videos

1 Introduction to Geometric Series with an example

\displaystyle \begin{array}{lll} &S_n = \frac{a(r^n - 1)}{r -1}, \text{ if } &|r| >1 \\ &S_n = \frac{a(1- r^n )}{1 - r}, \text{ if } &|r| <1 \end{array}

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1.54mins
1 Introduction to Geometric Series with an example

2 Geometric Series without using the Formula

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2.07mins
2 Geometric Series without using the Formula

7 Infinite repeating decimals as Geometric Series

\displaystyle \begin{array}{lll} &S_{\infty} = \frac{a}{1 - r}, \text{ if } &|n| <1 \end{array}

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3.35mins
7 Infinite repeating decimals as Geometric Series