Finite Differences

Example:

Find the sequence of forward differences for the sequence

$\begin{array}{l}{a}_n=n^2\\ 1,4,9,16,25,36,49,\mathrm{...}\end{array}$

Find the differences of consecutive terms.

$\begin{array}{l}4-1=3\\ 9-4=5\\ 16-9=7\\ 25-16=9\\ 36-25=11\\ 49-36=13\end{array}$

The sequence of forward differences is

$\begin{array}{l}3,5,7,9,11,13,\mathrm{...}\\ \Delta a_n=2n+1\end{array}$