Forward Difference of Power

Theorem

$\Delta c^x = \paren {c - 1} c^x$

where $\Delta$ denotes the forward difference operator.

Corollary

$\Delta 2^x = 2^x$

Proof

From the definitions:

\(\ds \Delta c^x\)

\(=\)

\(\ds c^{x + 1} - c^x\)

Definition of Forward Difference Operator

\(\ds \)

\(=\)

\(\ds c \cdot c^x - c^x\)

\(\ds \)

\(=\)

\(\ds \paren {c - 1} c^x\)

$\blacksquare$