Question

The sum of the first $n$ terms of an arithmetic progression is $S_n = 3n^2 + 2n$. What is the 5th term?

• A. 15
• B. 29
• C. 19
• D. 21

Hint:

For AP sum formulas, find the $n$-th term using $T_n = S_n - S_{n-1}$ and verify with the first few terms.

Answer 1

The Correct Option is B

- Step 1: The sum of the first $n$ terms is $S_n = 3n^2 + 2n$. The $n$-th term is $T_n = S_n - S_{n-1}$.
- Step 2: Calculate $S_{n-1}$: $S_{n-1} = 3(n-1)^2 + 2(n-1) = 3(n^2 - 2n + 1) + 2n - 2 = 3n^2 - 6n + 3 + 2n - 2 = 3n^2 - 4n + 1$.
- Step 3: Find $T_n$: $T_n = S_n - S_{n-1} = (3n^2 + 2n) - (3n^2 - 4n + 1) = 6n - 1$.
- Step 4: For the 5th term, $n = 5$: $T_5 = 6 \times 5 - 1 = 30 - 1 = 29$.
- Step 5: Check options: Try $T_5 = S_5 - S_4$. $S_5 = 3 \times 25 + 2 \times 5 = 85$, $S_4 = 3 \times 16 + 2 \times 4 = 56$,
so $T_5 = 85 - 56 = 29$.