* fix: clean-up Project Euler 121-140 * fix: corrections from review Co-authored-by: Sem Bauke <46919888+Sembauke@users.noreply.github.com> * fix: missing backticks Co-authored-by: Kristofer Koishigawa <scissorsneedfoodtoo@gmail.com> * fix: corrections from review Co-authored-by: Tom <20648924+moT01@users.noreply.github.com> * fix: missing delimiter Co-authored-by: Sem Bauke <46919888+Sembauke@users.noreply.github.com> Co-authored-by: Kristofer Koishigawa <scissorsneedfoodtoo@gmail.com> Co-authored-by: Tom <20648924+moT01@users.noreply.github.com>
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id, title, challengeType, forumTopicId, dashedName
id | title | challengeType | forumTopicId | dashedName |
---|---|---|---|---|
5900f3f51000cf542c50ff08 | Problem 137: Fibonacci golden nuggets | 5 | 301765 | problem-137-fibonacci-golden-nuggets |
--description--
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
.
For this problem we shall be interested in values of x
for which A_{F}(x)
is a positive integer.
Surprisingly
$$\begin{align} A_F(\frac{1}{2}) & = (\frac{1}{2}) × 1 + {(\frac{1}{2})}^2 × 1 + {(\frac{1}{2})}^3 × 2 + {(\frac{1}{2})}^4 × 3 + {(\frac{1}{2})}^5 × 5 + \cdots \\ & = \frac{1}{2} + \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \cdots \\ & = 2 \end{align}$$
The corresponding values of x
for the first five natural numbers are shown below.
x |
A_F(x) |
---|---|
\sqrt{2} − 1 |
1 |
\frac{1}{2} |
2 |
\frac{\sqrt{13} − 2}{3} |
3 |
\frac{\sqrt{89} − 5}{8} |
4 |
\frac{\sqrt{34} − 3}{5} |
5 |
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.
Find the 15th golden nugget.
--hints--
goldenNugget()
should return 1120149658760
.
assert.strictEqual(goldenNugget(), 1120149658760);
--seed--
--seed-contents--
function goldenNugget() {
return true;
}
goldenNugget();
--solutions--
// solution required