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

Nov 3, 2015

It could be ${a}_{n} = \frac{{n}^{3} + 5 n}{6}$

#### 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:

$\textcolor{b l u e}{1} , 3 , 7 , 14$

Write out the sequence of differences:

$\textcolor{b l u e}{2} , 4 , 7$

Write out the sequence of differences of those differences:

$\textcolor{b l u e}{2} , 3$

Write out the sequence of differences of those differences:

$\textcolor{b l u e}{1}$

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)

$= \textcolor{red}{\cancel{\textcolor{b l a c k}{1}}} + 2 n - \textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{{n}^{2}}}} - 3 n + \textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} + \frac{1}{6} {n}^{3} - \textcolor{red}{\cancel{\textcolor{b l a c k}{{n}^{2}}}} + \frac{11}{6} n - \textcolor{red}{\cancel{\textcolor{b l a c k}{1}}}$

$= \frac{{n}^{3} + 5 n}{6}$