Serie

s

Serie

s

1

5.6 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

Serie

s

Serie

s

2

5.6 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

Serie

s

Serie

s

3

5.6 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

Serie

s

Serie

s

4

5.6 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