Sequences and Series: Conceptual Doubts Cleared (4)

Question

The sum of the first $n$ terms of an AP is $3n^2 + 5n$. Find the $n$th term, and verify that the sequence is indeed arithmetic.

Solution — Step by Step

Final Answer: $a_n = 6n + 2$, with first term $8$ and common difference $6$.

Why This Works

For any sequence with cumulative sum $S_n$, the $n$th term is $S_n - S_{n-1}$ — this telescopes the cumulative sum back into individual terms. The sequence is an AP if and only if $S_n$ is a quadratic in $n$ with no constant term — and the coefficient of $n^2$ equals $d/2$.

Here $S_n = 3n^2 + 5n$ has no constant term, so it’s a clean AP. If the formula had been, say, $3n^2 + 5n + 7$, the sequence would still have $a_n = 6n + 2$ for $n \geq 2$ but a different $a_1$ (the "$+7$" disrupts the pattern).

Alternative Method

Compute $a_1 = S_1$ and $a_2 = S_2 - S_1$ directly, then $d = a_2 - a_1$, then $a_n = a_1 + (n-1)d$. Same answer, more arithmetic.