Indexed Summation without First Term

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

Let $a$ and $b$ be integers with $a \le b$.

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

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

Then we have an equality of indexed summations:


 * $\ds \sum_{i \mathop = a}^b \map f i = \map f a + \sum_{i \mathop = a + 1}^b \map f {\map \sigma i}$

Proof
The proof goes by induction on $b$.

Basis for the Induction
Let $b = a$.

We have:

This is our basis for the induction.

Induction Step
Let $b \ge a + 1$.

We have:

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

Also see

 * Indexed Summation over Adjacent Intervals