Question

The formula of the sum of first n natural numbers is $S={\displaystyle \frac{n\left(n+1\right)}{2}}$ . If the sum of first n natural number is 325 then find n.

Answer 1

Hint: In this question, we have a formula of sum of first n natural numbers and we know the sum of first n natural numbers given in question. So, put the value of sum in formula and get value of n after solving the quadratic equation.

Complete step-by-step answer:

We know the natural number form an A.P. and we have the sum of first n natural numbers is S=325.
Now, we apply the formula of sum of first n natural numbers mentioned in the question.
$$\begin{gathered} \Rightarrow S={\displaystyle \frac{n\left(n+1\right)}{2}} \\ \Rightarrow 325={\displaystyle \frac{n\left(n+1\right)}{2}} \\ \Rightarrow 650=n^2+n \\ \Rightarrow n^2+n-650=0 \end{gathered}$$
We can see quadratic equations in $n$ and solve the quadratic equation by using the Sridharacharya formula.
$$\begin{gathered} \Rightarrow n^2+n-650=0 \\ \Rightarrow n={\displaystyle \frac{-1\pm \sqrt{1-4\times 1\times \left(-650\right)}}{2}} \\ \Rightarrow n={\displaystyle \frac{-1\pm \sqrt{1+2600}}{2}} \\ \Rightarrow n={\displaystyle \frac{-1\pm \sqrt{2601}}{2}} \end{gathered}$$
Now, use $\sqrt{2601}=51$
$\Rightarrow n={\displaystyle \frac{-1\pm 51}{2}}$
We know the number of terms cannot be negative. So, we take only positive value.
$$\begin{gathered} \Rightarrow n={\displaystyle \frac{-1+51}{2}} \\ \Rightarrow n={\displaystyle \frac{50}{2}} \\ \Rightarrow n=25 \end{gathered}$$
So, the value of n is 25.

Note: Whenever we face such types of problems we use some important points. We can see the formula of sum of first n natural numbers mentioned in the question is the same as the formula of sum of first n terms of A.P. So, put the value of sum in formula then after some calculation we can get the required answer.