Let be the first term and the common difference of an arithmetic progression

MATHEMATICS · Subjective · CBSE · Mathematics · Arithmetic Progressions · Arithmetic Progressions

Problem

Let aa be the first term and dd the common difference of an arithmetic progression. Prove that for every positive integer nn, ∑k=0n−1(a+kd)=n2[2a+(n−1)d].\sum_{k=0}^{n-1}\bigl(a + k d\bigr) = \frac{n}{2}\left[2a + (n-1)d\right]. Provide a detailed proof based only on properties of sigma notation and standard summations.

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