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Consider the geometric series 4 + 20/a + 100/a^2 + .... If the sum is a perfect square, what is the smallest possible value of a where a is a positive integer?

 
 May 1, 2021
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Consider the geometric series 4 + 20/a + 100/a^2 + .... If the sum is a perfect square, what is the smallest possible value of a where a is a positive integer?

 

Using the geometric series formula, we have: 4/(1 - 5/a)

Simplifying we have: 4a/(a-5)

4a/(a-5) = k^2

4a = ak^2 - 5k^2

4a/k^2 = a-5

k^2 will always stay positive, so a-5 = 1 (least square)

a=6

laugh

 

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