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Middle Years

10.05 Geometric series (Enrichment)

Worksheet
Geometric series
1

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

a

u_1 = 2.187 and r = 1.134

b

u_1 = - 4.186 and r = - 2.848

2

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

a

Find the common ratio, r.

b

Find the sum of the first 9 terms, correct to one decimal place.

3

Find the sum of the first 12 terms in the series 4 + 12 + 36 + \ldots

4

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

5

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

6

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

Leave your answer correct to the nearest whole number.

7

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

Round your answer to three decimal places.

8

The first 3 terms of a sequence are x, \, \dfrac{3}{4} x^{2}, \, \dfrac{9}{16} x^{3}.

a

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

b

Evaluate the sum when x = 4 and n = 8.

9

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.

10

The sum of the first 7 terms of a geometric series is 2186, and the common ratio is \dfrac{1}{3}.

Find u_1, the first term in the series.

11

The sum of the first 6 terms of a geometric series is 9 times the sum of its first 3 terms.

Find r, the common ratio.

12

Find n, the number of terms of the following geometric series:

a

The sum of n terms in the geometric series 1 + 4 + 16 + \ldots is 89\,478\,485.

b

The sum of n terms in the geometric series 16 + 4 + 1 + \ldots is \dfrac{5461}{256}.

13

Consider the series 64 + 16 + 4 + \ldots + \dfrac{1}{1024}.

a

Find n, the number of terms in the series.

b

Find the sum of the series, correct to three decimal places.

14

Consider the series 4 - 12 + 36 - \ldots - 708\,588.

a

Find n, the number of terms in the series.

b

Find the sum of the series.

15

Consider the geometric series 20, - 4, \dfrac{4}{5}, \ldots

Find the number of terms n, that have a sum of \dfrac{10\,416}{625}.

16

Consider the series 5 + 10 + 20 + \ldots.

a

Find the sum of the first n terms of the series in terms of n.

b

Find n, the total number of terms if the sum is equal to 2555.

c

Find how many terms are less than 50\,000.

d

Find the first sum of n terms that is greater than 50\,000.

e

Prove that the sum of the first n terms is always 5 less than the \left( n + 1 \right)th term.

17

The powers of 3 greater than 1 form a geometric progression 3, 9, 27, \ldots

a
Find how many powers of 3 there are between 2 and 10^{25}.
b

Find how many terms must be added for the sum to exceed 10^{25}.

Applications
18

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?

19

Average annual salaries are expected to increase by 4 \% each year. If the average annual salary this year is found to be \$40\,000.

a

Calculate the expected average annual salary in 6 years.

b

This year, Vincent starts at a new job in which he will receive the average annual salary for each year of his employment. Over the coming 6 years (including this year) he plans to save half of each year’s annual salary.

Calculate his total savings over these 6 years.

20

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.

21

A conveyor belt is being used to remove materials from a quarry. Every 45 minutes, the conveyor belt empties out \dfrac{1}{5} of whatever material remains in the quarry. The quarry initially holds 19\,500 \text{ m}^{3} of materials.

a

How much material has been pumped out after 135 minutes?

b

How much material is left in the dam after 135 minutes?

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