Alia and The World Of Numbers

If n is the number of natural numbers then sum of natural numbers= n(n+1)/n

Now :- n(n+1)/2=120n

n(n+1)=240n

n^2 + n=240n

Divide by n on both sides then:-

n+1=240

Hence n=239

The Solution is Here : http://225.do.am/brilliant/alia.jpg

The sequence of numbers forms an arithmetic progression with a common difference of $1$ . The sum of the terms is given by $s=\dfrac{n}{2}(a_1+a_n)$ where $n$ is the number of terms, $a_1$ is the first term and $a_n$ is the $n^{th}$ term. Substituting, we have

$120n=\dfrac{n}{2}(1+n)$

$240=1+n$

$n=239$

Sum of $n$ natural numbers $= \frac{n(n+1)}{2}$

Thus, $\frac{n(n+1)}{2} = 120n$

$n(n+1)= 120n \times 2$

$n^{2}+n= 240n$

$n^{2}= 240n - n$

$n \times n= 239n$

$n= \frac{239n}{n}$

$n= 239$

Check: $\frac{239 \times (239+1)}{2} = 120 \times 239$

$239 \times 240 = 28680 \times 2$

$57360 = 57360$

So, $n = \boxed{239}$

Simple A.P