Find each sum:Find each sum:

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a) 3j=5

98

∑ b) (2k − 4)k=1

60

∑ c) n + 2( )!

n!n=3

5

∑

4, 12, 36, 108, . . .

A sequence is geometric if each term is obtained by multiplying the previous term by the same number called the common ratio, r.

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4, 4 ⋅3, 4 ⋅3⋅3, 4 ⋅3⋅3⋅3, . . .

The above sequence can be written as:

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a1, a2 , a3, a4 , . . . an

a1, a1 ⋅r1, a1 ⋅r

2 , a1 ⋅r3, . . . a1 ⋅r

n−1

Any term in a geometric sequence can be found using:

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an = a1 rn−1

Practice #6: p. 955 1-23odds

Ex. Find the common ratio, the fifth term, and the nth term formula for the following geometric sequence:

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28, − 7,74

, −7

16, . . .

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r = −14

a5 =7

64an = 28 −

14

⎛ ⎝ ⎜

⎞ ⎠ ⎟n−1

€

Partial Sum of a Geometric Sequence

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Sn = a1 +a1r +a1r2 + . . . a1r

n−1

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Snr = a1r +a1r2 +a1r

3 + . . .a1rn−1 + a1r

n

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Sn −Snr = a1 −a1rn

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Sn 1− r( ) = a1 1− rn( )

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Sn =a1 1− rn( )

1− r

Partial Sum of a Geometric Sequence

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sn =a1 1− rn( )

1− r

4, -12, 36, -108, . . .

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Ex. Find S3 for the above sequence.

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S3 =4 1− −3( )

3( )

1− −3( )( )

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=4 1+ 27( )

4

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=28 Practice #6: p. 955 25-35 odds

This is the second part of #6

Arithmetic, Geometric or Neither

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−3, 1, 5, 9 . . .

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−35, 5, −57

,549

. . .

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1, 4, 9, 16 . . .

€

Ex. Find a5 given the following : a1 = 3 and a3 =43

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43

= 3r 3−1

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43

= 3r2

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13⋅43

=13⋅3r2

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49

= r2

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±23

= r

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a5 = 323

⎛ ⎝ ⎜

⎞ ⎠ ⎟4

or a5 = 3 −23

⎛ ⎝ ⎜

⎞ ⎠ ⎟4

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a5 =1627

Bouncing Ball