Mathematics

Mathematics

Problem

Which of the following expressions represents the sum SnS_n of the first nn terms of an arithmetic progression with first term aa and common difference dd? a) Sn=n2 [a+(n−1)d]S_n=\dfrac{n}{2}\,[a+(n-1)d] b) Sn=n2 [2a+(n−1)d]S_n=\dfrac{n}{2}\,[2a+(n-1)d] c) Sn=n2 [2a+nd]S_n=\dfrac{n}{2}\,[2a+nd] d) Sn=a+n(n−1)2 dS_n=a+\dfrac{n(n-1)}{2}\,d
Intuitive explanation

An arithmetic progression grows by a constant amount each term, so you're really adding up a sequence where each term is aa, a+da+d, a+2da+2d, and so on. The key insight is that the first and last terms are symmetric around the middle—pair them up and you'll find they always sum to the same value. This lets you avoid adding all nn terms individually and instead use a clever counting argument based on how many pairs you have.

Intuitive explanation

An arithmetic progression grows by a constant amount each term, so you're really adding up a sequence where each term is aa, a+da+d, a+2da+2d, and so on. The key insight is that the first and last terms are symmetric around the middle—pair them up and you'll find they always sum to the same value. This lets you avoid adding all nn terms individually and instead use a clever counting argument based on how many pairs you have.

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