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 â€śOption dâ€ť.

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 donâ€™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.