63 lines
1.7 KiB
Markdown
63 lines
1.7 KiB
Markdown
---
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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 $A_{F}(x) = xF_1 + x^2F_2 + x^3F_3 + \ldots$, where $F_k$ is the $k$th term in the Fibonacci sequence: $1, 1, 2, 3, 5, 8, \ldots$; that is, $F_k = F_{k − 1} + F_{k − 2}, F_1 = 1$ and $F_2 = 1$.
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For this problem we shall be interested in values of $x$ for which $A_{F}(x)$ is a positive integer.
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Surprisingly
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$$\begin{align}
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A_F(\frac{1}{2}) & = (\frac{1}{2}) × 1 + {(\frac{1}{2})}^2 × 1 + {(\frac{1}{2})}^3 × 2 + {(\frac{1}{2})}^4 × 3 + {(\frac{1}{2})}^5 × 5 + \cdots \\\\
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& = \frac{1}{2} + \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \cdots \\\\
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& = 2
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\end{align}$$
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The corresponding values of $x$ for the first five natural numbers are shown below.
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| $x$ | $A_F(x)$ |
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|---------------------------|----------|
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| $\sqrt{2} − 1$ | $1$ |
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| $\frac{1}{2}$ | $2$ |
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| $\frac{\sqrt{13} − 2}{3}$ | $3$ |
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| $\frac{\sqrt{89} − 5}{8}$ | $4$ |
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| $\frac{\sqrt{34} − 3}{5}$ | $5$ |
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We shall call $A_F(x)$ a golden nugget if $x$ is rational, because they become increasingly rarer; for example, the 10th golden nugget is 74049690.
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Find the 15th golden nugget.
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# --hints--
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`goldenNugget()` should return `1120149658760`.
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```js
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assert.strictEqual(goldenNugget(), 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 goldenNugget() {
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return true;
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}
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goldenNugget();
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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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