Question

The sum $S$ of first $n$ natural numbers is given by the formula \[S = \dfrac{{n\left( {n + 1} \right)}}{2}\]. If \[S = 231\], find $n$.

Answer 1

Hint: Put the value of \[S = 231\] in the formula \[S = \dfrac{{n\left( {n + 1} \right)}}{2}\] and make a quadratic equation and find the values of $n$.

Complete step-by-step answer:

In the question it is already given that the sum $S$ of first $n$ natural numbers is given by the formula \[S = \dfrac{{n\left( {n + 1} \right)}}{2}\]. And we know that natural numbers are counting numbers starting from $1$ itself. Then to solve the question first of all put the value of \[S = 231\] such that
\[ \Rightarrow 231 = \dfrac{{n\left( {n + 1} \right)}}{2}\]
Now we will cross multiply the above equation as
\[ \Rightarrow 231 \times 2 = n\left( {n + 1} \right)\]
\[ \Rightarrow 462 = {n^2} + n\]
We now bring the terms to LHS i.e.
\[ \Rightarrow {n^2} + n - 462 = 0\]
This is a quadratic equation so we will solve it for $n$. On factorisation we get
\[ \Rightarrow {n^2} + 22n - 21n - 462 = 0\]
\[ \Rightarrow n\left( {n + 22} \right) - 21\left( {n + 22} \right) = 0\]
\[ \Rightarrow \left( {n + 22} \right)\left( {n - 21} \right) = 0\]
Equating both the factors to zero.
\[ \Rightarrow n = - 22\;\] Or \[n = 21\;\]
Therefore,\[n = 21\;\]as we know that \[n = - 22\;\] is not a natural number.

Note: Whenever this type of question appears then first all note down the given details as it gives the better understanding to solve the question. Remember that natural numbers are counting numbers starting from $1$ itself. Therefore, \[n = - 22\;\]is not a natural number.