Indexed Summation over Interval of Length Two

Theorem

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

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

Let $f: \set {a, a + 1} \to \mathbb A$ be a real-valued function.

Then the indexed summation:

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

Proof

We have:

\(\ds \sum_{i \mathop = a}^{a + 1} \map f i\)

\(=\)

\(\ds \sum_{i \mathop = a}^a \map f i + \map f {a + 1}\)

Definition of Indexed Summation

\(\ds \)

\(=\)

\(\ds \map f a + \map f {a + 1}\)

Indexed Summation over Interval of Length One

$\blacksquare$