GEOMETRIC SEQUENCES

These are sequences where the ratio of successive terms of a sequence is always the same number. This number is called the common ratio.

Notice in this sequence that if we find the ratio of any term to the term before it (divide them) we always get 2. 2 is then called the common ratio and is denoted with the letter r.

r = 2

To get to the next term in the sequence we would multiply by 2 so a recursive formula for this sequence is:

12 nn aa

1, 2, 4, 8, 16 . . .

2 2 2 2

Each time you want another term in the sequence you’d multiply by r. This would mean the second term was the first term times r. The third term is the first term multiplied by r multiplied by r (r squared). The fourth term is the first term multiplied by r multiplied by r multiplied by r (r cubed). So you can see to get the nth term we’d take the first term and multiply r (n - 1) times.

r = 2

1 nn ara

Try this to get the 5th term.

a = 1

1621 155 a

1, 2, 4, 8, 16 . . .

Let’s look at a formula for a geometric sequence and see what it tells us.

132 n

Subbing in the set of positive integers we get:

-2, -6, -18, -54 … What is the common ratio?

r = 3

you can see what the common ratio will be in the formula

This factor gets us started in the right place. With n = 1 we’d get -2 for the first term

3n-1 would generate the powers of 3. With the - 2 in front, the first term would be -2(30) =- 2. What would you do if you wanted the sequence -4, -12, -36, -108, . . .?

134 n

Find the nth term of the geometric sequence when a = -2 and r =4

If we use 4n-1 we will generate a sequence whose common ratio is 4, but this sequence starts at 1 (put 1 in for n to get first term to see this). We want ours to start at -2. We then need the “compensating factor”. We need to multiply by -2.

142 n

Check it out by putting in the first few positive integers and verifying that it generates our sequence.

-2, -8, -32, -128, . . .Sure enough---it starts at -2 and has a common ratio of 4

Find the 8th term of 0.4, 0.04. 0.004, . . .

1 nn ara

1.04.0

04.0r

To find the common ratio, take any term and divide it by the term in front

11.04.0 nna

00000004.01.04.0 188 a

If we want to add n terms in a geometric sequence, we use the formula below:

1,0,1

1

rr

raS

n

n

sum of n terms

first term

common ratio

number of terms

15

1

134n

nFind the sum:

4 + 12 + 36 + 108 + . . . + 4(3)14

31

314

15

15

S

= 28,697,812

Let’s look at the sum of the geometric sequence

n

na

2

1

16

1

8

1

4

1

2

1

0 1

Let’s look at this on the number line

2

1

4

1

8

1

16

1

Each time we add another term we’d be going half the distance left. As n the sum 1.

means infinity

If the common ratio was not a fraction between -1 and 1, then the sequence would keep getting larger and larger and would as n . If the common ratio is a fraction between -1 and 1, the sum as n is as follows:

r

aar

k

k

11

1first term

common ratio

Let’s try this for the previous sequence:

16

1

8

1

4

1

2

1

21

1

21

S 1

2121

2

1x

2

1x

2

1x

Let’s try one more:

1

1

3

18

k

k

r

aS

1

8

3

112

328

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au