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Middle Years

10.03 Geometric sequences

Worksheet
Geometric sequences
1

Write down the next two terms for the following sequences:

a

4, 12, 36, \ldots

b

12, - 48, 192, \ldots

c

- 1, 8, - 64, \ldots

d

- 6, 9, -\dfrac{27}{2},\ldots

2

Consider the sequence -1, -7, -49, \ldots

a

Find the next term of the sequence.

b

Find the 5th term of the sequence.

c

Find the 6th term of the sequence.

3

Explain how the common ratio of a geometric sequence can be found.

4

Suppose u_{1}, u_{2}, u_{3}, u_{4}, u_{5},\ldots is a geometric sequence.

Is u_{1}, u_{3}, u_{5},\ldots also a geometric sequence? Explain your answer.

5

Consider the first four terms of the following geometric sequences:

i

Evaluate \dfrac{u_2}{u_1}.

ii

Evaluate \dfrac{u_3}{u_2}.

iii

Evaluate \dfrac{u_4}{u_3}.

iv

Hence, find u_5.

a
- 8, - 16, - 32, - 64, \ldots
b

- 4 , - 8 , - 16 , - 32 , \ldots

c

2, - 6 , 18, - 54 , \ldots

d

- 64 , - 16 , - 4 , -1, \ldots

6

State the common ratio between the terms of the following sequences:

a

9, 36, 144, 576, \ldots

b

- 6 , - 42 , - 294 , - 2058 , \ldots

c

2, - 16, 128, - 1024, \ldots

d

- 70.4 , - 17.6 , - 4.4 , - 1.1 ,\ldots

7

Write the first 5 terms of the following sequences given the first term and the common ratio:

a

First term: - 2, common ratio: 3

b

First term: 1.3, common ratio: - 4.

c

First term: 700\,000, common ratio: 1.04.

8

For each of the following, write the first four terms in the geometric progression:

a

The first term is 6 and the common ratio is 4.

b

The first term is 7 and the common ratio is - 2.

c

The first term is 700\,000 and the common ratio is 1.04.

d

The first term is - 2 and the common ratio is 3.

e

The first term is 1.3 and the common ratio is - 4.

9

Find the missing terms in the following geometric progressions:

a

- 5, \, x, \, - 80, \, 320, \, y

b

a, \, b, \, \dfrac{3}{25}, \, - \dfrac{3}{125}, \, c

10

For each of the following pairs of terms in a geometric progression:

i

Find the possible values of r.

ii

Find the value of u_1.

iii

Find the general rule for u_n, for r \gt 0.

a

u_3 = 18 and u_5 = 162

b

u_4 = 32 and u_6 = 128

Applications
11

Suppose you save \$1 on the first day of a month, \$2 on the second day, \$4 on the third day, \$8 on the fourth day, and so on. That is, each day you save twice as much as you did the day before.

a

How much will you put aside for savings on the 6th day of the month?

b

How much will you put aside for savings on the 10th day of the month?

12

The average daily growth of a seedling is 10\% per day. A seedling measuring 6 \text{ cm} in height is planted.

a

Determine the height of the seedling at the end of Day 1.

b

Find the height of the seedling 2 days after it is planted.

c

Write a recursive rule for H_n, defining the height of the seedling n days after it is planted, and an initial condition H_0.

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