# Linear Combination of Indexed Summations

## Theorem

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

Let $a,b$ be integers.

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

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

Let $\lambda, \mu \in \mathbb A$.

Let $\lambda \cdot f + \mu \cdot g$ be the sum of the product of $f$ with $\lambda$ and the product of $g$ with $\mu$.

Then we have the equality of indexed summations:

$\displaystyle \sum_{i \mathop = a}^b \left( \lambda \cdot f(i) + \mu \cdot g(i) \right) = \lambda \cdot \sum_{i \mathop = a}^b f(i) + \mu \cdot \sum_{i \mathop = a}^b g(i)$

## Proof

We have:

 $\displaystyle \sum_{i \mathop = a}^b \left( \lambda \cdot f(i) + \mu \cdot g(i) \right)$ $=$ $\displaystyle \sum_{i \mathop = a}^b \left( \lambda \cdot f(i) \right) + \sum_{i \mathop = a}^b \left( \mu \cdot g(i) \right)$ Indexed Summation of Sum of Mappings $\displaystyle$ $=$ $\displaystyle \lambda \cdot \sum_{i \mathop = a}^b f(i) + \mu \cdot \sum_{i \mathop = a}^b g(i)$ Indexed Summation of Multiple of Mapping

$\blacksquare$