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 $a\leq b$.

Let $\left[{a \,.\,.\, b}\right]$ be the integer interval between $a$ and $b$.

Let $f : \left[{a \,.\,.\, b}\right] \to \mathbb A$ be a mapping.

Then we have an equality of indexed summations:


 * $\displaystyle \sum_{i \mathop = a}^b f(i) = f(a) + \sum_{i \mathop = a+1}^b f(\sigma(i))$

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 $b \geq a+1$.

We have:

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

Also see

 * Indexed Summation over Adjacent Intervals