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8.05 Geometric series

Worksheet
Geometric series
1

Find the sum of the first 5 terms of the series: 24 - 12 + 6 \ldots

Leave your answer correct to the nearest whole number.

2

Find the sum of the first 7 terms of the geometric series: 64 + 16 + 4 \ldots

3

Find the sum of the first 7 terms of the series: 8 + 1 + \dfrac{1}{8} \ldots

Round your answer to three decimal places.

4

Find the sum of the first 11 terms of the series: 1 - 2 + 4 \ldots

5

Find the sum of the first 12 terms of the series: 5 + 10 + 20 \ldots

6

Find the sum of the first 5 terms of the geometric sequence defined by the following. Round your answers to two decimal places.

a

t_1 = 2.187 and r = 1.134

b

t_1 = - 4.186 and r = - 2.848

7

For each of the following series:

i

Find n, the number of terms in the series.

ii

Find the sum of the series.

a
4 - 8 + 16 - \ldots - 2048
b
64 + 16 + 4 + \ldots + \dfrac{1}{16}
8

Find n, the number of terms, in each of the following series:

a

The sum of n terms in the geometric series 2 + 10 + 50 + \ldots is 195\,312.

b

The sum of n terms in the geometric series 5,- 20, 80,\ldots is 262\,145.

c

The sum of n terms in the geometric series 16 + 4 + 1 + \ldots is \dfrac{1365}{64}.

d

The sum of n terms in the geometric series 40,- 8,\dfrac{8}{5},\ldots is \dfrac{20\,832}{625}.

9

Consider the series: 5 + \dfrac{5}{2} + \dfrac{5}{4} \ldots

a

Find the common ratio, r.

b

Find the sum of the first 10 terms, rounding your answer to one decimal place.

10

The first 3 terms of a sequence are \, x, \, \dfrac{2}{5} x^{2}, \, \dfrac{4}{25} x^{3}.

a

Write a simplified expression for the sum of the first n terms of this sequence.

b

Evaluate the sum when x = 5 and n = 9.

11

Consider the series: \dfrac{1}{4} - \dfrac{1}{5} + \dfrac{1}{16} - \dfrac{1}{25} + \dfrac{1}{64} - \dfrac{1}{125} + \ldots

Form an expression for the sum of the first 2 n terms of the series.

12

The sum of the first 6 terms of a geometric series is 28 times the sum of its first 3 terms.

Find r, the common ratio.

13

The sum of the first 9 terms of a geometric series is 436\,905, and the common ratio is \dfrac{1}{4}.

Find t_1, the first term in the series.

Summation notation
14

Write the following series using summation notation:

a
1+2+4+ \ldots + 512
b

1 - \dfrac{1}{2} + \dfrac{1}{4} - \dfrac{1}{8} + \ldots + \dfrac{1}{1024}

c
-3+9-27+ \ldots +729
d
6+18+54+ \ldots + 486
e
\dfrac{1}{16}+\dfrac{1}{32}+\dfrac{1}{64}+ \ldots + \dfrac{1}{1024}
f
8+4+2+\ldots + \dfrac{1}{8}
15

Evaluate:

a
\sum_{i=1}^{4} 4^{i}
b
\sum_{j=1}^{5} 243 \left(\dfrac{5}{3}\right)^{j}
c
\sum_{j=1}^{5} 32 \left(\dfrac{1}{2}\right)^{j}
d
\sum_{i=3}^{8} \left( - 3 \right)^{i}
e
\sum_{i=2}^{7} \left( - 2^{i} \right)
f
\sum_{i=1}^{5} \left( 2 \left(3\right)^{i - 1}\right)
g
\sum_{j=4}^{8} \left( \dfrac{1}{4} \left(3\right)^{j - 1}\right)
h
\sum_{j=1}^{4} \left( - 3 \left(\dfrac{1}{3}\right)^{j} \right)
16

Write the following series using summation notation:

x+2x^4+4x^7+8x^{10} + \ldots 2^{18}x^{55}
Technology
17

Use the sequence facility of your calculator to determine for which term number n the geometric sequence T_n = 6 \left(2.1^{n - 1}\right) has a sum of 217.3146.

18

Use the sequence facility of your calculator to determine for which term number n the geometric sequence T_n = 800 \left(0.9^{n - 1}\right) has a sum of 4556.262\,32.

19

The 5th term of a geometric sequence is 405. The sum of the 6th, 7th and 8th terms is 15\,795.

a

Write an equation involving T_1, the first term, and r, the common ratio, of this geometric sequence for the 5th term.

b

Write an equation involving T_1, the first term, and r, the common ratio, and the sum of the 6th, 7th and 8th terms of the sequence.

c

Use the simultaneous solving facility of your calculator to find the values of T_1 and r. Assume T_1 and r are positive.

d

Hence, state the general term, T_n, of the sequence.

Applications
20

Suppose you save \$1 the first day of a month, \$2 the second day, \$4 the third day, \$8 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 your total savings be for the first 13 days?

b

How much will your total savings be for the first 29 days?

21

Average annual salaries are expected to increase by 5\% each year. The average annual salary this year is found to be \$49\,000.

a

Calculate the expected average annual salary in 4 years.

b

This year, Aaron starts at a new job in which he will receive the average annual salary for each year of his employment. Over the coming 4 years (including this year), he plans to save half of each year’s annual salary.

Determine his total savings over these 4 years.

22

This year, 600 people are expected to enter the workforce as registered nurses. This number is expected to increase by 4\% next year, and increase by the same percentage every year after that.

Calculate the following, rounding your answers to the nearest whole number:

a

The number of nurses expected to enter the workforce between six and seven years from now.

b

The number of nurses expected to enter the workforce over the next six years.

23

A conveyor belt is being used to remove materials from a quarry. Every thirty minutes, the conveyor belt empties out \dfrac {1}{5} of whatever material remains in the quarry.

The quarry initially holds 14\,000\text{ m}^3 of materials.

a

How much material has been taken out after 90 minutes?

b

How much material is left in the quarry after 90 minutes?

24

The first blow of a hammer drives a post a distance of 64\text{ cm} into the ground. Each successive blow drives the post \dfrac {3}{4} as far as the preceding blow. In order for the post to become stable, it needs to be driven \dfrac {781}{4}\text{ cm} into the ground.

If n is the number of hammer strikes needed for the pole to become stable, find n.

25

A car’s brakes failed and the driver immediately turned the engine of the car off. In the first second after the engine was shut off, the car travelled 20 \text{ m}. Every successive second, the car travelled 80\% of the distance covered in the previous second.

a

Find an expression for the total distance the car travelled in the first n seconds

b

Hence, find the total distance the car travelled in the first four seconds to two decimal places.

c

Traffic had built up 109 \text{ m} away from where the driver turned off the engine.

How far from the traffic build-up did the car come to rest?

26

At the start of 2014, Pauline deposits \$5000 into an investment account. At the end of each quarter, she makes an extra deposit of \$700.

The table below shows the first few quarters of 2014. All values in the table are in dollars:

QuarterOpening BalanceInterestDepositClosing Balance
\text{Jan - Mar}50002007005900
\text{Apr - Jun}5900236.007006836.00
\text{Jul - Sep}6836.00273.447007809.44
a

Find the quarterly interest rate.

b

Write an expression for the amount in the account at the end of the first quarter.

c

Hence, write an expression for the amount in the account at the end of the second quarter.

d

Hence, write a similar expression for the amount in the account at the end of the third quarter.

e

Write an expression for the amount in the investment account after n quarters.

f

Hence, determine the total amount in Pauline’s account at the beginning of 2016 to the nearest dollar.

27

Iain opens an account to help save for a house. He opens the account at the beginning of 2017 with an initial deposit of \$50\,000 that is compounded annually at a rate of 5.9\% per annum. He makes further deposits of \$3000 at the end of each year.

a

Write an expression for the amount in the account after:

i

1 year

ii

2 years

iii

3 years

b

The amount in the account after n years can be expressed as: 50\,000 \times \left(1.059\right)^{n} + 3000 \times \left(1.059\right)^{n - 1} + \ldots + 3000 \times \left(1.059\right)^{2} + 3000 \times 1.059 + 3000This can be written as 50\,000 \times \left(1.059\right)^{n}, plus the sum of a geometric sequence. Write an expression representing this sum.

c

Hence, determine the total value of Iain's savings in 2024, to the nearest dollar.

28

Lisa invests \$30\,000 at a rate of 1.5\% per month compounded monthly. Each month, she withdraws \$600 from her investment after the interest is paid and the balance is reinvested in the account.

a

Write an expression for A_{1}, the balance of the account after 1 month.

b

Write an expression for:

i

A_{2}

ii

A_{3}

c

Hence, write an expression for A_{n}.

d

Calculate the amount Lisa has saved after 3 years.

29

To save up to buy a car, Laura opens a savings account that earns 6\% per annum compounded monthly.

She initially deposits \$1400 when she opens the account at the beginning of the month, and then deposits \$165 at the end of every month.

a

The amount in the account after n months can be expressed as the nth term of a geometric sequence plus the sum of a different geometric sequence.

Write an expression for the amount in the investment account after n months.

b

Hence, determine the amount Laura has saved after 3 years.

30

Rochelle invests \$190\,000 at a rate of 7\% per annum compounded annually, and wants to work out how much she can withdraw each year to ensure the investment lasts 20 years.

a

Write an expression for A_{1}, the balance of the account after 1 year. Use x to represent the amount to be withdrawn each year.

b

Write an expression for:

i

A_{2}

ii

A_{3}

c

Hence, write an expression for A_{n}.

d

Hence, determine Rochelle's annual withdrawal amount.

31

Mario invests \$110\,000 at a rate of 16\% per annum compounded quarterly and wants the investment to last 25 years.

a

The amount in the account after n quarters can be expressed as the nth term of a geometric sequence minus the sum of a different geometric sequence.

Write an expression for the amount in the investment account after n quarters. Use x to represent the amount to be withdrawn each quarter.

b

Hence, determine Mario's quarterly withdrawal amount.

32

Emma invests \$190\,000 at a rate of 3\% per annum compounded monthly, and wants to work out how much she can withdraw each month to ensure the investment lasts 30 years.

a

The amount in the account after n months can be expressed as the nth term of a geometric sequence minus the sum of a different geometric sequence.

Write an expression for the amount in the investment account after n months. Use x to represent the amount to be withdrawn each month.

b

Hence, determine Emma's monthly withdrawal amount.

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Outcomes

2.2.8

establish and use the formula S_n=t_1 (r^n−1)/(r−1) for the sum of the first n terms of a geometric sequence

2.2.9

use geometric sequences in contexts involving geometric growth or decay, such as compound interest

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