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Geometric
Sequences & SeriesBy: Jeffrey Bivin
Lake Zurich High School
Last Updated: October 11, 2005
7/29/2019 Geometric Sequence Series
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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
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nth term
of geometric sequence
an = a1r(n-1)
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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)
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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
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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
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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
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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
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Sum it up
r
raaraS
nn
i
n
n
1
11
1
1
1
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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
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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
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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:
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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
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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
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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
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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)
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Geometric
Infinite Series
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The Magic Flea(magnified for easier viewing)
There is
no flea
like aMagic
Flea
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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...
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Sum it up -- Infinity
r
araS
i
n
1
1
1
1
1
1
rfor
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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
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311
6
S
96
32
S
r
aS
1
1
...27
2
9
2
3
226 S
61 a
31r
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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?
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A Bouncing Ball
Downward = 128 + 64 + 32 + 16 + 8 +
256128
1
128
1 21
2
1
1
r
aS
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A Bouncing Ball
Upward = 64 + 32 + 16 + 8 +
12864
1
64
1 21
2
1
1
r
aS
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A Bouncing Ball
Upward = 64 + 32 + 16 + 8 + = 128Downward = 128 + 64 + 32 + 16 + 8 + = 256
TOTAL = 384 ft.
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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?
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A Bouncing BallDownward = 625 + 375 + 225 + 135 + 81 +
5.1562625
1
625
152
53
1
r
aS
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A Bouncing Ball
Upward = 375 + 225 + 135 + 81 +
5.937375
1
375
152
53
1
r
aS
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A Bouncing Ball
Upward = 375 + 225 + 135 + 81 + = 937.5Downward = 625 + 375 + 225 + 135 + 81 + = 1562.5
TOTAL = 2500 ft.
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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
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Fractions - Decimals
911.
922.
31
933.
9
44.
955.
32
966.
97
7.
988.
19.99
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Lets try again
313.
3
1
3.
3
13.
13
39.
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One more
9.xlet
9.910 x
subtract 99 x1
99 x
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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
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Thats AllFolks
19.
.9 = 1.9 = 1