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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?

 Apr 13, 2022
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Sum of an infinite geometric  series  =     first term  /   [ 1  - common  ratio ]

 

The common ratio   =    5/a

 

So

 

4 / [ 1 - 5/a ]    =

 

4 / [ (a-5) / a ]    =

 

4a / [ a -5 ]

 

The smallest  positive a that makes the sum a perfect square is when a = 9 

 

4(9)  / [ 9 -5]  =  36 / 4  =   9

 

cool cool cool

 Apr 13, 2022

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