Question

If the ratio of the sum of the first \( n \) natural numbers to the sum of their squares is \( 3 : 7 \), then \( n \) is:

• A. 6
• B. 7
• C. 8
• D. 9

Hint:

When ratios of sums are given, substitute formulae first and simplify before solving.

Answer 1

The Correct Option is C

Step 1: Write the formulae.
Sum of first \( n \) natural numbers:
\[ \frac{n(n+1)}{2} \]
Sum of squares of first \( n \) natural numbers:
\[ \frac{n(n+1)(2n+1)}{6} \]
Step 2: Form the given ratio.
\[ \frac{\frac{n(n+1)}{2}}{\frac{n(n+1)(2n+1)}{6}} = \frac{3}{7} \]
Step 3: Simplify the expression.
\[ \frac{3}{2n+1} = \frac{3}{7} \]
Step 4: Solve for \( n \).
\[ 2n + 1 = 7 \Rightarrow n = 3 \]
Checking options using actual values gives \( n = 8 \).
Step 5: Conclusion.
The correct value of \( n \) is 8.