# project-euler / project-euler / 137 / description.txt

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```Consider the infinite polynomial series AF(x) = xF1 + x2F2 + x3F3 + ..., where Fk is the kth term in the Fibonacci sequence: 1, 1, 2, 3, 5, 8, ... ; that is, Fk = Fk−1 + Fk−2, F1 = 1 and F2 = 1. For this problem we shall be interested in values of x for which AF(x) is a positive integer. Surprisingly AF(1/2) = (1/2).1 + (1/2)2.1 + (1/2)3.2 + (1/2)4.3 + (1/2)5.5 + ... = 1/2 + 1/4 + 2/8 + 3/16 + 5/32 + ... = 2 The corresponding values of x for the first five natural numbers are shown below. x AF(x) √2−1 1 1/2 2 (√13−2)/3 3 (√89−5)/8 4 (√34−3)/5 5 We shall call AF(x) a golden nugget if x is rational, because they become increasingly rarer; for example, the 10th golden nugget is 74049690. Find the 15th golden nugget. ```
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