Question

A student of $5^{th}$ standard started writing down the counting numbers as $1,2,3,4,....$ and then he added all those numbers and got the result $500$. But when I checked the result I have found that he has missed a number. What is the missing number?

Answer 1

Given: $1+2+3+...+n=500$

$\Rightarrow \frac{n(n+1)}{2}=500$

$\Rightarrow n^2+n=1000$

$\approx n^2=1000$

Since ${31}^2<1000<{32}^2$

$\Rightarrow 961<1000<1024$

Now, the perfect square closest to $1000$ is $1024={32}^2$

Since the student didnot add one number, so $500$ is lesser than the actual sum and $1000$ is also less than the actual value.

Hence, we will take $32$

Hence, initially $32$ were added.

So their sum will be $\frac{32\times (32+1)}{2}=33\times 16=528$

As the given sum is $500$ so the student forgot to add $528-500=28$

$\therefore$ the missing number is $28$