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 $\closedint a c$ denote the integer interval between $a$ and $c$.

Let $b \in \closedint {a - 1} c$.

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

Then we have an equality of indexed summations:


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

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 $a \le b \le c$.

We have:

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

Also see

 * Sum over Disjoint Union of Finite Sets