Question

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

Accepted Answer

35. Sum preceding $x$ is $S_{x-1} = \frac{(x-1)x}{2}$. Sum following $x$ is $S_{49} - S_x = \frac{49 	imes 50}{2} - \frac{x(x+1)}{2} = 1225 - \frac{x(x+1)}{2}$. Equating both sums: $\frac{(x-1)x}{2} + \frac{x(x+1)}{2} = 1225 \implies \frac{2x^2}{2} = 1225 \implies x^2 = 1225 \implies x = 35$.

House numbers equal sum problem

Explanation

Practise this under board exam conditions