Forward Difference of Power
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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 | |||||||||||
\(\ds \) | \(=\) | \(\ds c \cdot c^x - c^x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {c - 1} c^x\) |
$\blacksquare$