Mathematics

Problem

In any arithmetic progression, the

n

th term

a_n

can be expressed using the partial sums

S_n

and

S_{n-1}

as a)

a_n = S_n + S_{n-1}

a_n = S_n - S_{n-1}

a_n = \dfrac{S_n}{S_{n-1}}

a_n = S_{n-1} - S_{n-2}

Intuitive explanation

The key insight is that any term in a sequence can be isolated by taking the difference between consecutive partial sums. Since $S_n$ represents the sum of all terms up to $a_n$ , and $S_{n-1}$ represents the sum up to $a_{n-1}$ , subtracting them leaves only $a_n$ behind. This relationship works for any sequence, not just arithmetic progressions, making it a powerful tool for extracting individual terms from cumulative information.

Intuitive explanation

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