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Geometric

Sequences & SeriesBy: Jeffrey Bivin

Lake Zurich High School

[email protected]

Last Updated: October 11, 2005

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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

5

1

1

14

3

2

3

23

n

n

n

n

Jeff Bivin -- LZHS

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nth term

of geometric sequence

an = a1r(n-1)

Jeff Bivin -- LZHS

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Find the nth term of the

geometric sequenceFirst term is 2Common ratio is 3

an = a1r(n-1)

an = 2(3)(n-1)

Jeff Bivin -- LZHS

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Find the nth term of a

geometric sequenceFirst term is 128Common ratio is (1/2)

1

2

1128

n

na

1

7

2

12

nna

an = a1r(n-1)

82

1

nn

a

Jeff Bivin -- LZHS

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Find the nth term of the

geometric sequenceFirst term is 64Common ratio is (3/2)

1

2

364

n

na

1

16

2

32

n

n

na

an = a1r(n-1)

7

1

2

3

n

n

na

Jeff Bivin -- LZHS

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Finding the 10th term

3, 6, 12, 24, 48, . . .a1 = 3

r = 2

n = 10

an = a1r(n-1)

an = 3(2)10-1an = 3(2)

9

an = 3(512)an = 1536

Jeff Bivin -- LZHS

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Finding the 8th term

2, -10, 50, -250, 1250, . . .a1 = 2

r = -5

n = 8

an = a1r(n-1)

an = 2(-5)8-1an = 2(-5)

7

an = 2(-78125)an = -156250

Jeff Bivin -- LZHS

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Sum it up

r

raaraS

nn

i

n

n

1

11

1

1

1

Jeff Bivin -- LZHS

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1 + 3 + 9 + 27 + 81 + 243

a1 = 1

r = 3

n = 6 313116

nS

2

7291

nS

364

2

728

nS

r

raaSn

n1

11

Jeff Bivin -- LZHS

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4 - 8 + 16 - 32 + 64 128 + 256

a1 = 4

r = -2

n = 7)2(1)2(447

nS

3

)128(44 nS

172

3

516

3

5124

nS

r

raaSn

n1

11

Jeff Bivin -- LZHS

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Alternative Sum Formula

r

raaS

n

n1

11

11

nn raa

rraran

n 1

1n

n rara 1

r

raaS nn

1

1

We know that:

Multiply by r:

Simplify:

Substitute:

Jeff Bivin -- LZHS

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Find the sum of the

geometric Series

3

21

3

2

729

256

5

n

S

3

1

2187

5125

n

S

729

10423nS

r

raaS nn

1

1

729

256na

3

2r

51 a

Jeff Bivin -- LZHS

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Evaluate

a1 = 2

r = 2n = 10

an = 10242122210

nS

1

102422

nS

2046

1

2046

nS

r

raaSn

n1

11

10

1

2

k

k= 2 + 4 + 8++1024

Jeff Bivin -- LZHS

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Evaluate

a1 = 3

r = 2n = 8

an = 384212338

nS

1

25633

nS

765

1

765

nS

r

raaSn

n1

11

8

1

123

j

j

= 3 + 6 + 12 ++ 384

Jeff Bivin -- LZHS

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Review -- Geometric

r

raa

S

n

n

1

1

nth term Sum of n terms

r

raa

S

n

n

111

an = a1r(n-1)

Jeff Bivin -- LZHS

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Geometric

Infinite Series

Jeff Bivin -- LZHS

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The Magic Flea(magnified for easier viewing)

There is

no flea

like aMagic

Flea

Jeff Bivin -- LZHS

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The Magic Flea(magnified for easier viewing)

1...32

1

16

1

8

1

4

1

2

1S

2

1

4

1

8

1

16

1

32

1...

Jeff Bivin -- LZHS

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Sum it up -- Infinity

r

araS

i

n

1

1

1

1

1

1

rfor

Jeff Bivin -- LZHS

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2

1

2

1

1

S

121

21

S

r

aS

1

1

21

1 a

2

1r

1...32

1

16

1

8

1

4

1

2

1

S

Remember --The Magic Flea

Jeff Bivin -- LZHS

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311

6

S

96

32

S

r

aS

1

1

...27

2

9

2

3

226 S

61 a

31r

Jeff Bivin -- LZHS

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A Bouncing Ballrebounds of the distance from which it fell --

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

from the top of a 128 feet tall building?

Jeff Bivin -- LZHS

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A Bouncing Ball

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

256128

1

128

1 21

2

1

1

r

aS

Jeff Bivin -- LZHS

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A Bouncing Ball

Upward = 64 + 32 + 16 + 8 +

12864

1

64

1 21

2

1

1

r

aS

Jeff Bivin -- LZHS

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A Bouncing Ball

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

TOTAL = 384 ft.

Jeff Bivin -- LZHS

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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?

Jeff Bivin -- LZHS

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

5.1562625

1

625

152

53

1

r

aS

Jeff Bivin -- LZHS

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A Bouncing Ball

Upward = 375 + 225 + 135 + 81 +

5.937375

1

375

152

53

1

r

aS

Jeff Bivin -- LZHS

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A Bouncing Ball

Upward = 375 + 225 + 135 + 81 + = 937.5Downward = 625 + 375 + 225 + 135 + 81 + = 1562.5

TOTAL = 2500 ft.

Jeff Bivin -- LZHS

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1.1

9.

S

19.

9.S

r

aS

1

1

...00009.0009.009.09.9. nS

9.1 a

1.r

Find the sum of the series

Jeff Bivin -- LZHS

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Fractions - Decimals

911.

922.

31

933.

9

44.

955.

32

966.

97

7.

988.

19.99

Jeff Bivin -- LZHS

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Lets try again

313.

3

1

3.

3

13.

13

39.

Jeff Bivin -- LZHS

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One more

9.xlet

9.910 x

subtract 99 x1

99 x

Jeff Bivin -- LZHS

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1.1

9.

S

1

9.

9.9. S

r

aS

1

1

...00009.0009.009.09.9.9.

9.1

a

1.r

OK now a series

Jeff Bivin -- LZHS

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Thats AllFolks

19.

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