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5.6 Sum of Geometric Series (1/4). a(r n -1) r-1. a(1-r n ) 1-r. S n = for |r|>1. S n = for |r|
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s5.6 Sum of Geometric Series (1/4)
SSnn = for |r|>1= for |r|>1a(ra(rnn-1)-1)r-1r-1
Sum (add) Sum (add) to n termsto n terms
TermTermNumberNumberFirstFirst
TermTermCommon
Ratio
SSnn = for |r|<1= for |r|<1a(1-ra(1-rnn))1-r1-r
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s5.6 Sum of Geometric Series (2/4)
Example: Example: Find the sum of the first 10 terms of Find the sum of the first 10 terms of
22 ++ 66 ++ 1818 + …+ …
rr = 6÷2 = 3 Sn = since |3|>1a(rn – 1)r - 1
aa = 2
nn = 10 S10 =2x(310 – 1)
3 - 1
= 59 048
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s5.6 Sum of Geometric Series (3/4)
Example: Example: Evaluate Evaluate 22 ++ 66 ++ 1818 + … + 486+ … + 486
Tn = a rn – 1
Find nFind n Find SFind Snn
rr = 6÷2 = 3aa = 2
TTnn = 486486 = 2x3n – 1
3n – 1 = 243
log33n – 1 = log3243n - 1 = log3243
n - 1 =ln 243ln 3
n – 1 = 5n = 6
Sn = since |3|>1a(rn – 1)r - 1
S6 =2x(36 – 1)
3 - 1
= 728
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s5.6 Sum of Geometric Series (4/4)
Example: Example:
EvaluateEvaluate
7
2
n3
7327
2
n 3...333 2187...279
a = 9 r = 3 n = 6
Sn = since |3|>1a(rn – 1)r - 1
S6 =9x(36 – 1)
3 - 1
= 9837
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