Let $\{a_{n}\}$ be a sequence of real numbers. The backward differences of this sequence are defined recursively as shown next. The first difference $\triangledown a_{n}$ is
$$\triangledown a_{n} = a_{n} − a_{n−1}.$$
The $(k + 1)^{\text{st}}$ difference $\triangledown^{k+1}a_{n}$ is obtained from $\triangledown ^{k} a_{n}$ by
$$\triangledown ^{k+1}a_{n} = \triangledown^{k}a_{n} − \triangledown ^{k}a_{n−1}.$$
Show that any recurrence relation for the sequence $\{a_{n}\}$ can be written in terms of $a_{n}, \triangledown a_{n}, \triangledown^{2}a_{n},\dots$ The resulting equation involving the sequences and its differences is called a difference equation.
