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

Let $f, g : \closedint a b \to \mathbb A$ 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:

$\ds \sum_{i \mathop = a}^b \paren {\lambda \cdot \map f i + \mu \cdot \map g i} = \lambda \cdot \sum_{i \mathop = a}^b \map f i + \mu \cdot \sum_{i \mathop = a}^b \map g i$

Proof

We have:

\(\ds \sum_{i \mathop = a}^b \paren {\lambda \cdot \map f i + \mu \cdot \map g i}\)

\(=\)

\(\ds \sum_{i \mathop = a}^b \paren {\lambda \cdot \map f i} + \sum_{i \mathop = a}^b \paren {\mu \cdot \map g i}\)

Indexed Summation of Sum of Mappings

\(\ds \)

\(=\)

\(\ds \lambda \cdot \sum_{i \mathop = a}^b \map f i + \mu \cdot \sum_{i \mathop = a}^b \map g i\)

Indexed Summation of Multiple of Mapping

$\blacksquare$

Also see

• Linear Combination of Convergent Series