Geometric

Sequences & Series

Geometric Sequences

1, 2, 4, 8, 16, 32, … 2n-1, …

3, 9, 27, 81, 243, … 3n, . . .

81, 54, 36, 24, 16, … , . . .1

3

281

n

nth term of geometric sequence

an = a1·r(n-1)

Find the nth term of thegeometric sequence

First term is 2

Common ratio is 3

an = a1·r(n-1)

an = 2(3)(n-1)

EX1 Find the nth term of a geometric sequence

a) First term is 128

Common ratio is (1/2)

1

2

1128

n

na

an = a1·r(n-1)

Ex 1 Find the nth term of the geometric sequence

b) First term is 64

Common ratio is (3/2)

1

2

364

n

na

1

16

2

32

n

n

na

an = a1·r(n-1)

7

1

2

3

n

n

na

c) Finding the 10th term3, 6, 12, 24, 48, . . .a1 = 3

r = 2

n = 10an = a1·r(n-1)

an = 3·(2)10-1

an = 3·(2)9

an = 3·(512)an = 1536

d) Finding the 8th term2, -10, 50, -250, 1250, . . .a1 = 2

r = -5

n = 8an = a1·r(n-1)

an = 2·(-5)8-1

an = 2·(-5)7

an = 2·(-78125)

an = -156250

Sum it up

11 1

1

1

1

nnn

ni

rS a r a

r

1 + 3 + 9 + 27 + 81 + 243

a1 = 1

r = 3

n = 6

61 311 3nS

3642

728

nS

1

1

1

n

n

rS a

r

EX 2 Find the suma) 4 - 8 + 16 - 32 + 64 – 128 + 256

a1 = 4

r = -2

n = 7

71 241 2nS

516

1723nS

1

1

1

n

n

rS a

r

b) Evaluate

a1 = 2

r = 2

n = 10

101 221 2nS

20461

2046

nS

1

1

1

n

n

rS a

r

10

1

2k

k= 2 + 4 + 8+…+1024

c) Evaluate

a1 = 3

r = 2

n = 8

81 231 2nS

7651

765

nS

1

1

1

n

n

rS a

r

8

1

123j

j= 3 + 6 + 12 +…+ 384

Review -- Geometric

nth term Sum of n terms

1

1

1

n

n

rS a

r

an = a1·r(n-1)

Geometric

Infinite Series

Sum it up -- Infinity

r

araS

i

n

1

1

1

11

1rfor

21

21

1S

121

21

S

r

aS

1

1

21

1 a

21r

1 1 1 1 1) ... 1

2 4 8 16 32a S

EX 3 Find the sum

311

6

S

96

32

S

r

aS

1

1

2 2 2) 6 2 ...

3 9 27b S

61 a

3

1r

c) A Bouncing Ballrebounds ½ of the distance from which it fell --

What is the total vertical distance that the ball traveled before coming to rest if it fell

from the top of a 128 feet tall building?

A Bouncing Ball

Downward = 128 + 64 + 32 + 16 + 8 + …

256128

1

128

1 21

21

1

r

aS

A Bouncing Ball

Upward = 64 + 32 + 16 + 8 + …

12864

1

64

1 21

21

1

r

aS

Jeff Bivin -- LZHS

A Bouncing Ball

Upward = 64 + 32 + 16 + 8 + … = 128

Downward = 128 + 64 + 32 + 16 + 8 + … = 256

TOTAL = 384 ft.

d) A Bouncing Ballrebounds 3/5 of the distance from which it fell --

What is the total vertical distance that the ball traveled before coming to rest if it fell from the top

of a 625 feet tall building?

A Bouncing BallDownward = 625 + 375 + 225 + 135 + 81 + …

5.1562625

1

625

1 52

53

1

r

aS

A Bouncing BallUpward = 375 + 225 + 135 + 81 + …

5.937375

1

375

1 52

53

1

r

aS

A Bouncing Ball

Upward = 375 + 225 + 135 + 81 + … = 937.5

Downward = 625 + 375 + 225 + 135 + 81 + … = 1562.5

TOTAL = 2500 ft.