Geometric Sequences & Series

This chapter focuses on how to use

find terms of a geometric sequence

or series, find the sum of finite and

infinite geometric series and apply

the rules for geometric sequences

and series to problems of growth

and decay.

Geometric Sequences & Series

CONTENTS:

What is a geometric sequence?

First term and common ratio

Example 1

Nth term of a sequence or series

Example 2

Example 3

Assignment

What is a Geometric Sequence or Series?

A sequence or series where we get from one

term to the next by multiplying the

previous term by a constant number is

called a geometric sequence or series.

These are examples of geometric sequences:

(i)3, 9, 27, 81, ... Multiply by 3

(ii)16, 8, 4, 2, 1, 0.5, ...Multiply by ½

(iii)1, -2, 4, -8, 16, ... Multiply by -2

Geometric Sequences & Series

First Term

The first of a geometric sequence or series is

usually called “a”.

Common Ratio

The constant number that each term is

multiplied by is called the common ratio.

To find the common ratio we take two terms

and divide one by the previous one.

Geometric Sequences & Series

Example 1:Find the common ratio of this sequence:100, 25, 6.25, 1.5625, ...

Solution:To get the common ratio we divide a term of the sequences by the previous term. If we take the first two terms of the above sequence we get:

25/100 = ¼

Therefore r, the common ratio is ¼

Geometric Sequences & Series

nth term of a Geometric Sequence or Series

If given the first term and the common ratio

we can find any term of a geometric

sequence or series by using the formula:

nth term = arn-1

Therefore,

First term = a Second term = ar

Third term = ar2 Fourth term = ar3 ...

Geometric Sequences & Series

Example 2:Find the 10th term of the following geometric sequence:2, 6, 18, 54, ...

Solution:Begin by writing down the values of a and r.a = 2 r = 6/2 = 3

To get any term we substitute into the general formula: arn-1 where n is the number of the term we want.

10th term: ar10-1 = ar9 = 2(3)9 = 39366

Geometric Sequences & Series

Example 3: The second term of a geometric sequence is 4 and the 4th term is 16. Find:(a) common ratio (b) first term

Solution:In this question we are trying to find two unknown variables, r and a. If we have two unknown variables we need to have two equations to solve.

Equation 1: 2nd term = 4This gives: ar = 4

continued on next slide

Geometric Sequences & Series

Solution continued:Equation 2:4th term = 16This gives: ar3 = 16

Because our equations involve terms multiplied together we need to divide the equations to eliminate one of the variables.

Divide (2) by (1) to get:ar3/ar = 16/4r2 = 4r = 2

continued on next slide

Geometric Sequences & Series

Solution continued:We can find the value of a by substituting the value of r into either equation. Substituting into Equation 1 gives:

a(2) = 4a = 2

Geometric Sequences & Series

AssignmentThis weeks assignment is a Moodle Activity. There are 5 questions to answer. Please be sure to include your working out for all questions.

Deadline is 5:00pm on Monday 15th March.

Geometric Sequences & Series