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Solutions
45 Videos

Which is a recursion formula for the sequence shown?

**A** `f(n) = f(n -1) + 4`

**B** `f(n) = 4n -2`

**C** `f(n) = 2 + (n -1)(4)`

**D** `f(1) = 2, f(n) = f(n-1) + 4 `

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0.22mins

Q1

Which expressions represent the missing terms in the binomial expansion shown?

```
\displaystyle
(x + y)^7 = x^7 + 7x^6y + ... + 35x^4y^ + 35x^3 y^4 + 21x^2y^5+ ... + y^7
```

**A** `21y^5x^2, 7yx^6`

**B** `21x^5y^2, 7xy^6`

**C** `-21x^5y^2, - 7xy^6`

**D** `x^5y^2, xy^6`

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1.03mins

Q2

What is the formula for the general term of an arithmetic sequence with `a = 8`

and `d=2`

?

**A** `t_n = 2+ (n + 1)(8)`

**B** `8 + (n -1)(2)`

**C** `8 + (n + 1)(2)`

**D** `2 + (n - 1)(8)`

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0.27mins

Q3

What are the first three terms of a geometric sequence with `a = 3`

and `r = 2`

?

**A** `3, 5, 7`

**B** `2, 6, 18`

**C** `3, 6, 12`

**D** `2, 5, 8`

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0.19mins

Q4

Which series is neither arithmetic nor geometric?

**A** `5+15+21+27+...`

**B** `1+8+27+64+...`

**C** `64-32+16+-8+...`

**D** `-3-2.7-2.4-2.1-...`

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0.57mins

Q5

Determine the first five terms of the sequence. Determine if the sequence is arithmetic, geometric or neither.

`t_n = 9 - 5n`

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0.27mins

Q6a

Determine the first five terms of the sequence. Determine if the sequence is arithmetic, geometric or neither.

`f(n) = 2n^2 + 3n -4`

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1.56mins

Q6b

Determine the first five terms of the sequence. Determine if the sequence is arithmetic, geometric or neither.

`f(n) = \frac{1}{8}(4)^{n -1}`

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0.48mins

Q6c

`t_n = 0.2n + 0.8`

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0.37mins

Q6d

`t_n = \frac{n + 4}{2}`

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0.52mins

Q6e

`f(n) = -3(2)^n`

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0.46mins

Q6f

Write an explicit formula and recursive formula for the sequence.

`64, 32, 16, 8, ...`

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0.22mins

Q7a

Write an explicit formula and recursive formula for the sequence.

`\displaystyle -20, -17, -14, -11, ...`

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0.19mins

Q7b

Write an explicit formula and recursive formula for the sequence.

`\displaystyle -4000, 1000, -250, 62.5, ...`

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0.18mins

Q7c

Write an explicit formula and recursive formula for the sequence.

`64, 32, 16, 8, ...`

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0.27mins

Q7d

Write an explicit formula and recursive formula for the sequence.

`\displaystyle -3, -6, -12, -24, ...`

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0.19mins

Q7e

Write an explicit formula and recursive formula for the sequence.

`\displaystyle -12\sqrt{2}, -10\sqrt{2}, -8\sqrt{2}, -6\sqrt{2}, ...`

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0.42mins

Q7f

Write `t_{11}`

for each sequence.

`6, 10, 14, 18, ...`

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0.26mins

Q8a

Write `t_{11}`

for each sequence.

`-3, -6, -12, -24, ...`

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0.22mins

Q8b

Write `t_{11}`

for each sequence.

`5, -10, 20, -40, ...`

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0.29mins

Q8c

Write `t_{11}`

for each sequence.

`-5, -10, -15, -20, ...`

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0.32mins

Q8d

Given the explicit formula, write `t_{15}`

for the sequence.

`\displaystyle f(n) = 2(-3)^{n + 1}`

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0.20mins

Q9a

Given the explicit formula, write `t_{15}`

for the sequence.

`\displaystyle t_n = 25n + 50`

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0.12mins

Q9b

Given the explicit formula, write `t_{15}`

for the sequence.

`\displaystyle t_n = 25n + 50`

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0.17mins

Q9c

Given the explicit formula, write `t_{15}`

for the sequence.

`\displaystyle f(n) = \frac{-3n}{4}`

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0.17mins

Q9d

Determine the number of terms in each sequence.

`5, 8, 11, ..., 62`

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0.42mins

Q10a

Determine the number of terms in each sequence.

`-4, 12, -36, ..., -19 131 876`

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1.20mins

Q10b

A new lake is being excavated. One day, 1.6 t of material is removed from the lake bed. On each of 10 days after that, `5`

% more is removed.

(a) Write the first three excavation amounts as a sequence.

(b) Write a recursion formula to represent the amount removed each day. Use this to determine the amount removed on the fifth day.

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1.45mins

Q11

Determine the specified sum for the series.

`S_{10}`

for `200 + 100 + 50 + ...`

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1.03mins

Q12a

Determine the specified sum for the series.

`S_{18}`

for `12+ 5 -2 + ...$``

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0.50mins

Q12b

Determine the specified sum for the series.

`120 + 110 + 100+ ... - 250`

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1.20mins

Q13a

Determine the specified sum for the series.

`8 + 24 + 40 + ... + 280`

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1.22mins

Q13b

Determine the sum of the geometric series.

`\displaystyle \frac{2}{81} + \frac{4}{27} + \frac{8}{9}+ ... + 6912`

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1.47mins

Q14a

Determine the sum of the geometric series.

`\displaystyle 5 + 10 + 20 + ... + 2560`

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0.56mins

Q14b

Use Pascal's triangle to help you expand the expression.

`(b - 3)^5`

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0.57mins

Q15a

Use Pascal's triangle to help you expand the expression.

`(2x - 5y)^6`

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1.10mins

Q15b

The sum of the first three terms of a series is 32. Determine the fourth term if the sum of the first four terms is `40`

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0.49mins

Q16a

The sum of the first three terms of a series is `32`

. Determine the fourth term if the sum of the first four terms is `25`

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0.19mins

Q16b

Determine the sum of the first 15 terms of an arithmetic series if the middle term is 92.

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1.56mins

Q17

Which is greater, A or B? Explain your reasoning.

`A = 50^2 - 49^2 + 48^2 - 47^2 + ... + 2^2 - 1^2`

`B = 50 +49 + 48+ 47+ ... + 2 + 1`

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0.57mins

Q18

In the arrangement of letters shown, starting from the top, proceed to the row below by moving diagonally to the immediate right or left. Determine the number of different paths that will spell the name PASCAL.

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0.56mins

Q19

A new wood stain loses 6.5% of its colour every year in a city that experiences a lot of hot, sunny days. What percent of colour will a fence in this city have 6 years after being stained?

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1.08mins

Q20

In an arithmetic series, the 4th term is 62 and the 14th term is 122. Determine the sum of the first 30 terms.

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1.20mins

Q21

A sailboat worth $140 000 depreciates 18% in the first year and 10% every year after that. How much will it be worth 8 years after it is bought?

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1.02mins

Q22

A magic square is an arrangement of numbers in which all rows, columns, and diagonals have the same sum. Using the magic square shown, substitute each number with the corresponding term from the Fibonacci sequence.

Show that the sum of the products of the rows is equal to the sum of the products of the columns.

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1.20mins

Q23