Sequences and Series: Common Mistakes and Fixes (5)

Question

The sum of the first $n$ terms of an AP is $S_n = 3n^2 + 5n$. Find (a) the first term, (b) the common difference, and (c) the 20th term.

Solution — Step by Step

Why This Works

The relation $a_n = S_n - S_{n-1}$ recovers the $n$-th term from the cumulative sum. For the first term, we use $a_1 = S_1$ directly. The common difference is then just $a_2 - a_1$.

If $S_n$ is a quadratic in $n$, the sequence is an AP — the coefficient of $n^2$ in $S_n$ equals $d/2$, and the linear coefficient relates to $a_1 - d/2$. This shortcut works only for APs.

Alternative Method

Match coefficients directly. For an AP, $S_n = (n/2)(2a + (n-1)d) = (d/2)n^2 + (a - d/2)n$. Comparing with $3n^2 + 5n$: $d/2 = 3$ so $d = 6$; $a - d/2 = 5$ so $a = 8$. Faster, no $S_2$ computation needed.

Common Mistake