To find the value of 'n' in this problem, we can follow these steps:
1. Let's denote the sum of the first 'n' natural numbers as \( S_n \).
2. The sum of the squares of the first 'n' natural numbers is given by \( S_{n^2} = 1^2 + 2^2 + 3^2 + ... + n^2 = \frac{n(n + 1)(2n + 1)}{6} \).
3. The square of the sum of the first 'n' natural numbers is \( S_n^2 = (1 + 2 + 3 + ... + n)^2 = \left(\frac{n(n + 1)}{2}\right)^2 \).
4. According to the problem, the ratio of \( S_{n^2} \) to \( S_n^2 \) is 25:234.
5. Set up the equation based on the ratio: \( \frac{S_{n^2}}{S_n^2} = \frac{n(n + 1)(2n + 1)/6}{(n(n + 1)/2)^2} = \frac{25}{234} \).
6. Solve for 'n' from the equation obtained in step 5.
7. After solving the equation, you will find the value of 'n'.
By following these steps and performing the calculations, you will determine the value of 'n' in this specific scenario.
