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. $1,8,48$
• B. $1,9,48$
• C. $1,9,49$
• D. $1,8,49$

Answer 1

The correct option is D $1,8,49$

Sum of first n natural numbers

$S=\frac{\left(n\right)(n+1)}{2}=$ perfect square

in options possible value of

$n=1,8,9,48,49$

$S_1=\frac{1\times 2}{2}=1=1^2\therefore n=1$

$S_8=\frac{8\times 9}{2}=36=6^2\Rightarrow n=6$

$S_9=\frac{9\times 10}{2}=45$

$S_{48}=\frac{48\times 49}{2}=24\times 49$

$S_{49}=\frac{49\times 50}{2}=49\times 25=(7\times 5{)}^2$$\Rightarrow n=49$

$\therefore n=1,8,49$ Answer (D)