What is the formula to this math sequence: 1, 3, 7, 14?

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What is the formula to this math sequence: 1, 3, 7, 14?

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Answer 1 · 2015-11-03T13:31:36

It could be $a_{n} = \frac{n^{3} + 5 n}{6}$ $a_n = (n^3+5n)/6$

Explanation:

You can always find a polynomial that matches a finite sequence like this one, but there are infinitely many possibilities.

Write out the original sequence:

$1, 3, 7, 14$ $color(blue)(1),3,7,14$

Write out the sequence of differences:

$2, 4, 7$ $color(blue)(2),4,7$

Write out the sequence of differences of those differences:

$2, 3$ $color(blue)(2),3$

Write out the sequence of differences of those differences:

$1$ $color(blue)(1)$

Having reached a constant sequence (!), we can write out a formula for $a_{n}$ $a_n$ using the first element of each sequence as a coefficient:

$a_{n} = \frac{1}{0!} + \frac{2}{1!} (n - 1) + \frac{2}{2!} (n - 1) (n - 2) + \frac{1}{3!} (n - 1) (n - 2) (n - 3)$ $a_n = color(blue)(1)/(0!)+color(blue)(2)/(1!)(n-1)+color(blue)(2)/(2!)(n-1)(n-2)+color(blue)(1)/(3!)(n-1)(n-2)(n-3)$

$=color(red)(cancel(color(black)(1)))+2n-color(red)(cancel(color(black)(2)))+color(red)(cancel(color(black)(n^2)))-3n+color(red)(cancel(color(black)(2)))+1/6n^3-color(red)(cancel(color(black)(n^2)))+11/6n-color(red)(cancel(color(black)(1)))$

$=(n^3+5n)/6$

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What is the formula to this math sequence: 1, 3, 7, 14?

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