# Moderate Ratios & Proportion Solved QuestionAptitude Discussion

 Q. 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:
 ✖ A. 15 ✔ B. 25 ✖ C. 35 ✖ D. 40

Solution:
Option(B) is correct

Sum of squares of first $n$ natural numbers

$=\sum{n^2}=$ $\dfrac{n(n+1)(2n+1)}{6}$

Squares of sum of first $n$ natural numbers

$= (\sum{n})^2=$$\dfrac{n(n+1)}{2}\times \dfrac{n(n+1)}{2}$

Now the ratio is

$\dfrac{n(n+1)(2n+1)}{6}$ : $\dfrac{n(n+1)}{2}\times \dfrac{n(n+1)}{2}$$=17:325$

$\Rightarrow \dfrac{\left(\frac{2n+1}{3}\right)}{\left(\frac{n(n+1)}{2}\right)}=\dfrac{17}{325}$

$\Rightarrow \dfrac{2(2n+1)}{3n(n+1)}=\dfrac{17}{325}$

$\Rightarrow 650(2n+1)=51n(n+1)$

$\Rightarrow 1300n+650=51n^2+51n$

$\Rightarrow 51n^2-1300n+51n-650=0$

$\Rightarrow 51n^2-1249n-650=0$

Upon solving the above equation, we get,

$n=25, -0.5098$

Out of these two, $n=\textbf{25}$ is there in the solution.

Thus option (B) is the right choice.

## (1) Comment(s)

Pranay
()

We can also approach from options.

Squares of sum of first n natural numbers (n(n+1)/2)^2 must be the multiple of 325 as the ratio given 17:325.

So by choosing each option we will get that only for 25 i.e. (25(25+1))^2 is divisible by 325.

ANS. 25