Question

Let \( S_n \) be the sum of the first \( n \) terms of an A.P. If \( S_n = 3n^2 + 5n \), then the sum of the squares of the first 10 terms of the given A.P. is:

• A. 15220
• B. 14220
• C. 15320
• D. 15110

Hint:

For \( S_n = An^2 + Bn \), the general term is always \( a_n = 2An + (B-A) \). Here \( A=3, B=5 \), so \( a_n = 6n + 2 \).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To find the terms, use \( a_n = S_n - S_{n-1} \). Once the general term is found, we square it and sum from \( n=1 \) to 10.

Step 2: Key Formula or Approach:
1. \( a_n = 6n + 2 \) (derived from \( S_n \)). 2. \( \sum_{n=1}^{10} a_n^2 = \sum_{n=1}^{10} (6n + 2)^2 \).

Step 3: Detailed Explanation:
1. \( (6n+2)^2 = 36n^2 + 24n + 4 \). 2. \( \sum 36n^2 = 36 \times \frac{10(11)(21)}{6} = 6 \times 10 \times 11 \times 7 \times 3 = 13860 \). 3. \( \sum 24n = 24 \times \frac{10(11)}{2} = 12 \times 110 = 1320 \). 4. \( \sum 4 = 4 \times 10 = 40 \). 5. Total = \( 13860 + 1320 + 40 = 15220 \).

Step 4: Final Answer:
The sum is 15220.