Question

The houses of a row are numbered 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

Answer 1

Here, we are given `S_(x-1) = S_49 - S_(x-1)-x`
`2S_(x-1)+x = S_49->(1)`
Here, `S_(x-1)` is the sum of first (x-1) natural numbers. `:. S_(x-1) = ((x-1)(x-1+1))/2 = (x(x-1))/2`
`S_49` is sum of first 49 natural numbers.
`:. S_49 = (49**50)/2 `
Putting these calues in (1), `2((x(x-1))/2)+x = (49**50)/2`
`=>x^2-x+x = 49**25`
`=> x = 7**5 =>x = 35`