• # question_answer 27) 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 proceeding the house numbered X is equal to sum of the numbers of houses following X.

 Given, the houses in a row numbered consecutively from 1 to 49. Now, sum of numbers preceding the number X $=\frac{X(X-1)}{2}$ And, sum of numbers following the number X $=\frac{49(50)}{2}-\frac{X(X-1)}{2}-X$ $=\frac{2450-{{X}^{2}}+X-2X}{2}$ $=\frac{2450-{{X}^{2}}-X}{2}$ According to the given condition, Sum of no's preceding X = Sum of no's following X $\frac{X(X-1)}{2}=\frac{2450-{{X}^{2}}-X}{2}$ ${{X}^{2}}-X=2450-{{X}^{2}}-X$ $2{{X}^{2}}=2450$ ${{X}^{2}}=1225$ $X=35$ Hence, at $X=35$, sum of no. of houses preceding the house no. X is equal to sum of the no. of houses following X.