SOLUTION: Calculate the sum of the series 4 + 12 + 36 + ... + 2916.

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Question 1202440: Calculate the sum of the series 4 + 12 + 36 + ... + 2916.


Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
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Calculate the sum of the series 4 + 12 + 36 + ... + 2916.
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The sequense  4, 12, 36, . . ., 2916  is a geometric progression
with the first term  a= 4  and the common ratio of r= 3  
(sinse 12/4 = 3 and the ratio of each next term to the current term is 3).


Find the number of term. Use the formula for the n-th term

    2916 = 4%2A3%5E%28n-1%29.


It gives  2916/4 = 729 = 3%5E%28n-1%29.  But 729 = 7%5E6;  therefore, n-1 = 7 and n= 7.

CHECK.  4%2A3%5E%287-1%29 = use your calculator = 2916,  correct.


To find the sum of this GP, use the general formula for the sum of a GP

    S%5Bn%5D = a%2A%28%28r%5En-1%29%2F%28r-1%29%29.


It gives at n= 7

    S%5B7%5D = 4%2A%28%283%5E7-1%29%2F%283-1%29%29 = %284%2F2%29%2A%283%5E7-1%29 = 2%2A%283%5E7-1%29 = use your calculator = 4372.   ANSWER

Solved.

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On geometric progressions,  see introductory lessons
    - Geometric progressions
    - The proofs of the formulas for geometric progressions
    - Problems on geometric progressions
    - Word problems on geometric progressions
in this site.


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