If Sum of First n Terms of AP is 3n² + 5n, Find nth Term

Question

The sum of the first $n$ terms of an AP is given by $S_n = 3n^2 + 5n$. Find the $n$th term of the AP.

Solution — Step by Step

Why This Works

Every time $S_n$ is given as a formula, the formula encodes the entire sequence. When we peel off $S_{n-1}$ from $S_n$, we’re literally removing the first $(n-1)$ terms, leaving only $a_n$.

Notice that $a_n = 6n + 2$ is a linear function of $n$ — that’s always the case when $S_n$ is quadratic. A quadratic $S_n$ generates a linear $a_n$, which is exactly what an AP should look like (constant common difference).

We can double-check the common difference: $d = a_2 - a_1 = 14 - 8 = 6$. Alternatively, the coefficient of $n$ in $a_n = 6n + 2$ directly gives $d = 6$. That’s a quick sanity check worth remembering.

Alternative Method

Once you suspect the AP structure, find $a_1$ and $d$ directly from $S_n$.

For any AP, there’s a standard result: $S_n = \frac{n}{2}[2a + (n-1)d]$. Expanding this gives $S_n = nd + n \cdot \frac{2a - d}{2}$… which gets messy. The cleaner shortcut:

Common Mistake