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?
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
