Mathematics

The key insight is that $S_n$ and $S_{n-1}$ represent cumulative sums that differ by exactly one term. When you subtract the sum of the first $n-1$ terms from the sum of the first $n$ terms, all the earlier terms cancel out, leaving only the $n$-th term itself. This is a direct way to extract any single term from an arithmetic sequence using the sum formula.

The key insight is that $S_n$ and $S_{n-1}$ represent cumulative sums that differ by exactly one term. When you subtract the sum of the first $n-1$ terms from the sum of the first $n$ terms, all the earlier terms cancel out, leaving only the $n$-th term itself. This is a direct way to extract any single term from an arithmetic sequence using the sum formula.