Question

Let \(S_n\) denote the sum of the squares of the first \(n\) odd natural numbers. If \(S_n = 533n\), find the value of \(n\).

• A. 18
• B. 20
• C. 24
• D. 30

Hint:

Use identity: \(\sum_{k=1}^{n} (2k - 1)^2 = \frac{n(4n^2 - 1)}{3}\) to evaluate sums of squares of odd numbers directly.

Answer 1

The Correct Option is B

Step 1: Use formula for sum of squares of first \(n\) odd numbers The formula is: \[ S_n = \sum_{k=1}^{n} (2k - 1)^2 = n(4n^2 - 1)/3 \] Use known formula: \[ \sum_{k=1}^{n} (2k - 1)^2 = \frac{n(4n^2 - 1)}{3} \] We are given: \[ \frac{n(4n^2 - 1)}{3} = 533n \Rightarrow \frac{4n^2 - 1}{3} = 533 \Rightarrow 4n^2 - 1 = 1599 \Rightarrow 4n^2 = 1600 \Rightarrow n^2 = 400 \Rightarrow n = \boxed{20} \] % Final Answer \[ \boxed{n = 20} \] % Correct Answer (updated): (b) 20