9.1 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.

Do you remember what an arithmetic sequence is?What formula is used?

An introduction…………

1, 4, 7,10,13

9,1, 7, 15

6.2, 6.6, 7, 7.4

, 3, 6

Arithmetic Sequences

ADDTo get next term

2, 4, 8,16, 32

9, 3,1, 1/ 3

1,1/ 4,1/16,1/ 64

, 2.5 , 6.25

Geometric Sequences

MULTIPLYTo get next term

Ex: Determine if the sequence is geometric. If so, identify the common ratio

• 1, -6, 36, -216

yes. Common ratio=-6

• 2, 4, 6, 8

no. No common ratio

This is an Arithmetic Sequence with “common difference” of 2

Important Formula for Geometric Sequence:

an = a1 r n-1

Where:

an is the nth term in the sequence

a1 is the first term

n is the number of the term

r is the common ratio

Ex: Write the first 4 terms of this sequence with:

First term: a1 = 7

Common ratio = 1/3

an = a1 * r n-1

Now find the first five terms:a1 = 7(1/3) (1-1) = 7a2 = 7(1/3) (2-1) = 7/3a3 = 7(1/3) (3-1) = 7/9a4 = 7(1/3) (4-1) = 7/27a5 = 7(1/3) (5-1) = 7/81

Geometric Sequence Problem

Find the 19th term in the sequence of 11,33,99,297 . . .

a19 = 11(3)18 =4,261,626,379

Common ratio = 3

a19 = 11 (3) (19-1)

Start with the sequence formula

Find the common ratio between the values.

Plug in known values

Simplify

an = a1 * r n-1

Let’s try one

Find the 10th term in the sequence of 1, -6, 36, -216 . . .

a10 = 1(-6)9 = -10,077,696

Common ratio = -6

a10 = 1 (-6) (10-1)

Start with the sequence formula

Find the common ratio between the values.

Plug in known values

Simplify

an = a1 * r n-1

2 2 2 2 r = 2

1 nn ara

Try this to get the 5th term.

a = 1

1621 155 a

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

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

Find the next four terms of –9, -2, 5, …

Arithmetic Sequence

2 9 5 2 7

7 is referred to as the common difference (d)

Common Difference (d) – what we ADD to get next term

Next four terms……12, 19, 26, 33

Find the next four terms of 0, 7, 14, …

Arithmetic Sequence, d = 7

21, 28, 35, 42

Find the next four terms of x, 2x, 3x, …

Arithmetic Sequence, d = x

4x, 5x, 6x, 7x

Find the next four terms of 5k, -k, -7k, …

Arithmetic Sequence, d = -6k

-13k, -19k, -25k, -32k

Given an arithmetic sequence with 15 1a 38 and d 3, find a .

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

x

15

38

NA

-3

n 1a a n 1 d

38 x 1 15 3

X = 80

16 1Find a if a 1.5 and d 0.5 Try this one:

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

1.5

16

x

NA

0.5

n 1a a n 1 d

16 1.5 0.a 16 51

16a 9

n 1Find n if a 633, a 9, and d 24

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

9

x

633

NA

24

n 1a a n 1 d

633 9 21x 4

633 9 2 244x

X = 27

1 29Find d if a 6 and a 20

1a First term

na nth term

nS sum of n terms

n number of terms

d common difference

-6

29

20

NA

x

n 1a a n 1 d

120 6 29 x

26 28x

13x

14

Example 7. An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. How many seats are there in all 20 rows?

1 20 1 19d c

1 201 20 19 1 39na a n d a

20

2020 39 10 59 590

2S

Example 8. A small business sells $10,000 worth of sports memorabilia during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 19 years. Assuming that the goal is met, find the total sales during the first 20 years this business is in operation.

1 10,000 7500 10,000 7500 2500a d c

1 201 10,000 19 7500 152,500na a n d a

20

2010,000 152,500 10 162,500 1,625,000

2S

So the total sales for the first 2o years is $1,625,000

Find the next three terms of 2, 3, 9/2, ___, ___, ___

3 – 2 vs. 9/2 – 3… not arithmetic3 9 / 2 3

1.5 geometric r2 3 2

3 3 3 3 3 3

2 2 2

92, 3, , , ,

2

9 9 9

2 2 2 2 2 2

92, 3, , ,

27 81 243

4 8,

2 16

1 9

1 2If a , r , find a .

2 3

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

1/2

x

9

NA

2/3

n 1n 1a a r

9 11 2

x2 3

8

8

2x

2 3

7

8

2

3 128

6561

9Find a of 2, 2, 2 2,...

1a First term

na nth term

nS sum of n terms

n number of terms

r common ratio

x

9

NA

2

2 2 2r 2

22

n 1n 1a a r

9 1

x 2 2

8

x 2 2

x 16 2