9.3 Geometric Sequences and Series

Common Ratio

In the sequence

2, 10, 50, 250, 1250, …..

Find the common ratio n

n

a

ar 1

5

510

50

550

250

r

r

r

Rule of Geometric Sequences

Let a1 = 2

r = 5

Find a6

1

1

n

n raa

6250

52

6

16

6

a

a

Rule of Geometric Sequences

Let a2 = 12

a3 = 36

Find a6

1

1

n

n raa

31236 r

Rule of Geometric Sequences

Let a2 = 12

a3 = 36

r = 3

Find a6

1

1

n

n raa

1

1

2

1

13

13

4936

336

336

a

a

a

aa

9722434

34

6

16

6

a

a

Finite Sum

The equation to find the Finite Sum of a Geometric Sequence

rr

araorSnn

k

k

n

11

11

1

1

Finite Sum

Let a1 = 7, r = 3.

Find the sum of the first 9 terms

rr

araorSnn

k

k

n

11

11

1

1

2

1968317

31

31737

99

1

1

k

k

Finite Sum

Let a1 = 7, r = 3.

Find the sum of the first 9 terms

rr

araorSnn

k

k

n

11

11

1

1

887,68984172

196827

2

1968317

31

31737

99

1

1

k

k

How many terms

0.25 + 0.75 + 2.25 + …….. = 820

Common ratio

rr

arann

k

k

1

11

1

1

1

325.0

75.0

375.0

25.2

r

r

How many terms

0.25 + 0.75 + 2.25 + …….. = 820

Common ratio 3

rr

arann

k

k

1

11

1

1

1

n

n

n

n

n

36561

316560

318

1820

2

31

4

1820

31

3125.0820

How many terms

0.25 + 0.75 + 2.25 + …….. = 820

Common ratio 3

rr

arann

k

k

1

11

1

1

1

n

n

n

n

n

36561

316560

318

1820

2

31

4

1820

31

3125.0820

8

3log6561log

6561log

36561

3

n

n

n

n

Infinite Geometric Series

If |r| < 1, it works

r

ara

n

n

11

1

1

1

Infinite Geometric Series

Since |0.2| < 1, it works. Let a1 = 42

5.528.042

2.0142

2.0421

1

n

n

Homework

Page 640 – 641

# 5, 15, 25, 35,

45, 55, 65, 75,

85, 105, 115

Homework

Page 640 – 641

# 3, 9, 27, 81, 113