Question

The houses in a row are numbered consecutively from $1$ to $49$. Show that there exists a value of X such that sum of numbers of houses preceding the house numbered X is equal to sum of the numbers of houses following X.

Answer 1

Formula,

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

Given,

$S_{x-1}=S_{49}-S_x$

$\frac{(x-1)\left(\right(x-1)+1)}{2}=\frac{49(49+1)}{2}-\frac{x(x+1)}{2}$

$x(x-1)=49\left(50\right)-x(x+1)$

$x^2-x=2450-x^2-x$

$2x^2=2450$

$x^2=1225$

$\therefore x=35$

Therefore number of houses is 35.