79d9012432
* fix(curriculum): tests quotes * fix(curriculum): fill seed-teardown * fix(curriculum): fix tests and remove unneeded seed-teardown
76 lines
2.3 KiB
Markdown
76 lines
2.3 KiB
Markdown
---
|
||
id: 5900f3d21000cf542c50fee4
|
||
challengeType: 5
|
||
title: 'Problem 101: Optimum polynomial'
|
||
---
|
||
|
||
## Description
|
||
<section id='description'>
|
||
If we are presented with the first k terms of a sequence it is impossible to say with certainty the value of the next term, as there are infinitely many polynomial functions that can model the sequence.
|
||
As an example, let us consider the sequence of cube numbers. This is defined by the generating function, un = n3: 1, 8, 27, 64, 125, 216, ...
|
||
Suppose we were only given the first two terms of this sequence. Working on the principle that "simple is best" we should assume a linear relationship and predict the next term to be 15 (common difference 7). Even if we were presented with the first three terms, by the same principle of simplicity, a quadratic relationship should be assumed.
|
||
We shall define OP(k, n) to be the nth term of the optimum polynomial generating function for the first k terms of a sequence. It should be clear that OP(k, n) will accurately generate the terms of the sequence for n ≤ k, and potentially the first incorrect term (FIT) will be OP(k, k+1); in which case we shall call it a bad OP (BOP).
|
||
As a basis, if we were only given the first term of sequence, it would be most sensible to assume constancy; that is, for n ≥ 2, OP(1, n) = u1.
|
||
Hence we obtain the following OPs for the cubic sequence:
|
||
|
||
OP(1, n) = 1
|
||
1, 1, 1, 1, ...
|
||
OP(2, n) = 7n−6
|
||
1, 8, 15, ...
|
||
OP(3, n) = 6n2−11n+6
|
||
1, 8, 27, 58, ...
|
||
OP(4, n) = n3
|
||
1, 8, 27, 64, 125, ...
|
||
|
||
Clearly no BOPs exist for k ≥ 4.
|
||
By considering the sum of FITs generated by the BOPs (indicated in red above), we obtain 1 + 15 + 58 = 74.
|
||
Consider the following tenth degree polynomial generating function:
|
||
un = 1 − n + n2 − n3 + n4 − n5 + n6 − n7 + n8 − n9 + n10
|
||
Find the sum of FITs for the BOPs.
|
||
</section>
|
||
|
||
## Instructions
|
||
<section id='instructions'>
|
||
|
||
</section>
|
||
|
||
## Tests
|
||
<section id='tests'>
|
||
|
||
```yml
|
||
tests:
|
||
- text: <code>euler101()</code> should return 37076114526.
|
||
testString: assert.strictEqual(euler101(), 37076114526, '<code>euler101()</code> should return 37076114526.');
|
||
|
||
```
|
||
|
||
</section>
|
||
|
||
## Challenge Seed
|
||
<section id='challengeSeed'>
|
||
|
||
<div id='js-seed'>
|
||
|
||
```js
|
||
function euler101() {
|
||
// Good luck!
|
||
return true;
|
||
}
|
||
|
||
euler101();
|
||
```
|
||
|
||
</div>
|
||
|
||
|
||
|
||
</section>
|
||
|
||
## Solution
|
||
<section id='solution'>
|
||
|
||
```js
|
||
// solution required
|
||
```
|
||
</section>
|