# Finite Differences

Given a sequence of numbers, we can find the sequence of forward differences of the sequence by subtracting adjacent terms.

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Example:
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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.

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}4-1=3\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}9-4=5\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}16-9=7\\ 25-16=9\\ 36-25=11\\ 49-36=13\end{array}$

The sequence of forward differences is

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

In general, if the original sequence is arithmetic , then the sequence of forward differences is constant; if the original sequence is quadratic, then the sequence of forward differences is arithmetic; and if the original sequence is geometric , then the sequence of forward differences is also geometric.