Geometric Sequence Series

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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.

    .9 = 1.9 = 1