Indexed Summation over Interval of Length One

Theorem

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

Let $a \in \Z$ be an integer.

Let $f: \set a \to \mathbb A$ be a mapping on the singleton $\set a$.


Then the indexed summation:

$\ds \sum_{i \mathop = a}^a \map f i = \map f a$


Proof

We have:

\(\ds \sum_{i \mathop = a}^a \map f i\) \(=\) \(\ds \sum_{i \mathop = a}^{a - 1} \map f i + \map f a\) Definition of Indexed Summation
\(\ds \) \(=\) \(\ds 0 + \map f a\) Definition of Indexed Summation, $a - 1 < a$
\(\ds \) \(=\) \(\ds \map f a\) Identity Element of Addition on Numbers

$\blacksquare$


Also see

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