Question

The houses of a row in a colony are numbered consecutively from 1 to 49. show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it . Find the value of x.

Answer 1

House numbers:  1 , 2, 3, 4, .....x-1, x , x+1, x+2, ..... 48, 49

Sum 1 : 1+2+3+...+x-1  = (x-1) x /2

Sum 2:  x+1,  x+2,  ..... 48+ 49
        This is an AP.  Number of terms: 49-x-1+1 = 49-x
         Sum2 = (49-x) (x+1 + 49)/2 

Sum1 = Sum2    =>  x(x-1) = (49-x)(x+50)
                                  2x² = 49 * 50
  x = 35