Indexed Summation of Multiple of Mapping

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

Let $a, b$ be integers.

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

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

Let $\lambda \in \mathbb A$.

Let $g = \lambda \cdot f$ be the product of $f$ with $\lambda$.

Then we have the equality of indexed summations:
 * $\ds \sum_{i \mathop = a}^b \map g i = \lambda \cdot \sum_{i \mathop = a}^b \map f i$

Proof
The proof goes by induction on $b$.

Basis for the Induction
Let $b < a$.

Then all indexed summations are zero.

Because $0 = \lambda \cdot 0$, the result follows.

This is our basis for the induction.

Induction Step
Let $b \geq a$.

We have:

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

Also see

 * Linear Combination of Indexed Summations
 * Summation of Multiple of Mapping on Finite Set
 * General Distributivity Theorem