Question

# If the sum of the first n natural numbers is a perfect square and n is less than 50, the possible values of n are:a. 1, 18, 48b. 1, 9, 48c. 1, 9, 49d. 1, 8, 49

Hint: Here, we have to equate the values given in the options in the formula of the sum of n natural numbers, which is given as $\dfrac{n\left( n+1 \right)}{2}$ and check if the resulting number is a perfect square or not.

We know that, the sum of n natural numbers is given by the formula, $\dfrac{n\left( n+1 \right)}{2}.........(i)$
So, now we will take all the possible values from the options given in the question.
So, the values of n are: 1, 8, 9, 18, 48 and 49.
Now, we will substitute each of these values in the formula, $\dfrac{n\left( n+1 \right)}{2}$ and then check, whether they result in a perfect square or not.
So, let us substitute the value, n = 1 in equation (i), so we get the sum as,
$\dfrac{n\left( n+1 \right)}{2}=\dfrac{1\left( 1+1 \right)}{2}$
On simplifying it further, we will get,
$\dfrac{1\left( 2 \right)}{2}=\dfrac{2}{2}=1$
Now, we will put n = 8 in equation (i), so we get the sum as,
$\dfrac{n\left( n+1 \right)}{2}=\dfrac{8\left( 8+1 \right)}{2}$
On further simplification, we will get,
$\dfrac{8\left( 9 \right)}{2}=\dfrac{72}{2}=36$
Now, on substituting n = 9 in equation (i), we get the sum as,
$\dfrac{n\left( n+1 \right)}{2}=\dfrac{9\left( 9+1 \right)}{2}$
On simplifying further, we get,
$\dfrac{9\left( 10 \right)}{2}=\dfrac{90}{2}=45$
Now, on substituting n = 48 in equation (i), we get the sum as,
$\dfrac{n\left( n+1 \right)}{2}=\dfrac{48\left( 48+1 \right)}{2}$
On simplifying further, we get,
$\dfrac{48\left( 49 \right)}{2}=\dfrac{2352}{2}=1176$
Now, on substituting n = 49 in equation (i), we get the sum as,
$\dfrac{n\left( n+1 \right)}{2}=\dfrac{49\left( 49+1 \right)}{2}$
On simplifying further, we get,
$\dfrac{49\left( 50 \right)}{2}=\dfrac{2450}{2}=1225$
Now, we will check, which of the resulting sums are a perfect square,
So, we get that, 1 is a perfect square as ${{\left( 1 \right)}^{2}}=1$.
Then 36 is a perfect square as ${{\left( 6 \right)}^{2}}=36$.
Also, we get 1225 is a perfect square as ${{\left( 35 \right)}^{2}}=1225$.
So, from the above observations, we can conclude that the correct answer is (1, 8, 49).

So, the correct answer is “Option d”.

Note: In order to solve this question quickly, the students must know a few perfect squares, like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, these are the squares of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 respectively. Otherwise, in order to check if a number is a perfect square or not, we will have to use the prime factorization method, which will be time consuming.
If the students don’t recollect the perfect squares, they can use the prime factorization method. So, in our solution, we can check if the number 1225 is a perfect square or not using the prime factorization method. So, we get,
\begin{align} & 5\left| \!{\underline {\, 1225 \,}} \right. \\ & 5\left| \!{\underline {\, 245 \,}} \right. \\ & 7\left| \!{\underline {\, 49 \,}} \right. \\ & 7\left| \!{\underline {\, 7 \,}} \right. \\ & \text{ }1 \\ \end{align}
So, on pairing the factors, we get, $\overline{5\times 5}\times \overline{7\times 7}$. So, we can write, $5\times 7=35$. Hence, we get that 1225 is a perfect square of the number 35.