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Question

Answers

a. 1, 18, 48

b. 1, 9, 48

c. 1, 9, 49

d. 1, 8, 49

Answer
Verified

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).

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.