55 lines
1.2 KiB
Markdown
55 lines
1.2 KiB
Markdown
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---
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id: 5900f3f51000cf542c50ff08
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title: 'Problem 137: Fibonacci golden nuggets'
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challengeType: 5
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forumTopicId: 301765
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dashedName: problem-137-fibonacci-golden-nuggets
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---
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# --description--
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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.
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For this problem we shall be interested in values of x for which AF(x) is a positive integer.
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Surprisingly AF(1/2)
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=
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(1/2).1 + (1/2)2.1 + (1/2)3.2 + (1/2)4.3 + (1/2)5.5 + ...
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= 1/2 + 1/4 + 2/8 + 3/16 + 5/32 + ...
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= 2 The corresponding values of x for the first five natural numbers are shown below.
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xAF(x) √2−11 1/22 (√13−2)/33 (√89−5)/84 (√34−3)/55
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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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# --hints--
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`euler137()` should return 1120149658760.
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```js
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assert.strictEqual(euler137(), 1120149658760);
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```
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# --seed--
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## --seed-contents--
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```js
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function euler137() {
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return true;
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}
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euler137();
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```
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# --solutions--
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```js
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// solution required
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```
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