Question

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

Answer 1

It could be `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:

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

Write out the sequence of differences:

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

Write out the sequence of differences of those differences:

`color(blue)(2),3`

Write out the sequence of differences of those differences:

`color(blue)(1)`

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

`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`