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---
id: 5900f3f51000cf542c50ff08
challengeType: 5
title: 'Problem 137: Fibonacci golden nuggets'
---
## Description
< section id = 'description' >
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.
xAF(x)
√2− 11
1/22
(√13− 2)/33
(√89− 5)/84
(√34− 3)/55
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.
< / section >
## Instructions
< section id = 'instructions' >
< / section >
## Tests
< section id = 'tests' >
```yml
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tests:
- text: < code > euler137()</ code > should return 1120149658760.
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testString: assert.strictEqual(euler137(), 1120149658760, '< code > euler137()< / code > should return 1120149658760.');
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```
< / section >
## Challenge Seed
< section id = 'challengeSeed' >
< div id = 'js-seed' >
```js
function euler137() {
// Good luck!
return true;
}
euler137();
```
< / div >
< / section >
## Solution
< section id = 'solution' >
```js
// solution required
```
< / section >