Indexed Summation over Translated Interval

Theorem
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$.

Let $a$ and $b$ be integers.

Let $\closedint a b$ be the integer interval between $a$ and $b$.

Let $f: \closedint a b \to \mathbb A$ be a mapping.

Let $c\in\Z$ be an integer.

Then we have an equality of indexed summations:


 * $\displaystyle \sum_{i \mathop = a}^b f(i) = \sum_{i \mathop = a + c}^{b + c} \map f {i - c}$

Proof
The proof goes by induction on $b$.

Basis for the Induction
Let $b < a$.

Then:
 * $b + c < a + c$

Thus both indexed summations are zero.

This is our basis for the induction.

Induction Step
Let $b \ge a$.

We have:

By the Principle of Mathematical Induction, the proof is complete.

Also see

 * Change of Variables in Indexed Summation