Indexed Summation over Adjacent Intervals

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

Let $a,b,c$ be integers.

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

Let $b \in \left[{a-1 \,.\,.\, c}\right]$.

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

Then we have an equality of indexed summations:


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

Outline of Proof
We proceed by induction, using Indexed Summation without First Term in the induction step.

Proof
The proof goes by induction on $b$.

Basis for the Induction
Let $b = a-1$.

We have:

This is our basis for the induction.

Induction Step
Let $a \leq b \leq c$.

We have:

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

Also see

 * Sum over Disjoint Union of Finite Sets