Sum of the first n terms of an AP

One of the most elegant properties of arithmetic progressions is that the sum of their first terms closes into a simple formula.

Property — Sum of the first $n$ terms of an AP

The sum of the first $n$ terms of an AP closes into a very clean formula: $S_n = a_1 + a_2 + \cdots + a_n = \frac{n\,(a_1+a_n)}{2} = \frac{n\bigl(2a_1+(n-1)d\bigr)}{2}.$ In words: “number of terms times the average of the first and last”.

The idea behind the formula is to pair the first term with the last, the second with the second-to-last, and so on: each pair has the same sum $a_1+a_n$, and there are $n/2$ pairs.

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Topics: Sequences
Concepts: Arithmetic progression
Methods: Sum of an arithmetic progression
Skills: Prove · Use formulas