Mathematics

You have two unknowns ($n$ and the common difference $d$) and two equations: one from the $n$-th term formula and one from the sum formula. The key insight is that the sum of an arithmetic progression depends on both the first and last terms—so use $S_n = \frac{n(a_1 + a_n)}{2}$ to connect what you know directly, then solve for $n$ without finding $d$ first.

You have two unknowns ($n$ and the common difference $d$) and two equations: one from the $n$-th term formula and one from the sum formula. The key insight is that the sum of an arithmetic progression depends on both the first and last terms—so use $S_n = \frac{n(a_1 + a_n)}{2}$ to connect what you know directly, then solve for $n$ without finding $d$ first.