The ratio of squares of first n natural numbers to square of sum of first n natural numbers is `17 : 325`. The value of n is :

Options

A. 15
B. 25
C. 35
D. 30

Answer

To solve the problem, we need to find the value of \( n \) given the ratio of the square of the sum of the first \( n \) natural numbers to the square of the sum of the first \( n \) natural numbers is \( 17 : 325 \).

### Step-by-step Solution:

1. **Understanding the formulas**:
- The sum of the first \( n \) natural numbers is given by:
\[ S_n = \frac{n(n + 1)}{2} \]
- The sum of the squares of the first \( n \) natural numbers is given by:
\[ S_{n, \text{squares}} = \frac{n(n + 1)(2n + 1)}{6} \]

2. **Setting up the ratio**:
- We need to find the ratio of the sum of squares to the square of the sum:
\[ \frac{S_{n, \text{squares}}}{(S_n)^2} = \frac{\frac{n(n + 1)(2n + 1)}{6}}{\left(\frac{n(n + 1)}{2}\right)^2} \]

3. **Simplifying the expression**:
- First, calculate \( (S_n)^2 \):
\[ (S_n)^2 = \left(\frac{n(n + 1)}{2}\right)^2 = \frac{n^2(n + 1)^2}{4} \]
- Now substituting into the ratio:
\[ \frac{\frac{n(n + 1)(2n + 1)}{6}}{\frac{n^2(n + 1)^2}{4}} = \frac{n(n + 1)(2n + 1)}{6} \cdot \frac{4}{n^2(n + 1)^2} \]
- This simplifies to:
\[ \frac{4(2n + 1)}{6n(n + 1)} = \frac{2(2n + 1)}{3n(n + 1)} \]

4. **Setting the ratio equal to \( \frac{17}{325} \)**:
- We set up the equation:
\[ \frac{2(2n + 1)}{3n(n + 1)} = \frac{17}{325} \]

5. **Cross-multiplying to eliminate the fraction**:
- Cross-multiplying gives:
\[ 2(2n + 1) \cdot 325 = 17 \cdot 3n(n + 1) \]
- This simplifies to:
\[ 650(2n + 1) = 51n(n + 1) \]

6. **Expanding both sides**:
- Expanding gives:
\[ 1300n + 650 = 51n^2 + 51n \]

7. **Rearranging the equation**:
- Rearranging leads to:
\[ 51n^2 - 1249n - 650 = 0 \]

8. **Using the quadratic formula**:
- We can use the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 51 \), \( b = -1249 \), and \( c = -650 \).
- Calculate the discriminant:
\[ b^2 - 4ac = (-1249)^2 - 4 \cdot 51 \cdot (-650) \]
- Solve for \( n \).

9. **Finding the value of \( n \)**:
- After calculating, we find that \( n = 25 \).

### Final Answer:
The value of \( n \) is \( 25 \).