# The houses of a row 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 values of x.

Clearly, No of houses are in AP and

Given,

First term, a = 1

common difference d = 1

According to given statement: S1 to x- 1 = S(x+1) to 49
Sx+1 to 49 = S49 – Sx and S1 to x- 1 = Sx- 1

Since the sum of n terms is Sn =

Where,

a = First term of AP
d = Common difference of AP
and no of terms is ‘n’

Applying given condition:

S1 to x- 1 = S(x+1) to 49
Sx- 1 = S49 – Sx
Sx- 1 = S49 – Sx

(x - 1)(x) = 49(50) - x(x+1)
x2 – x = 49(50) – x2 -x
2x2 = 49 × 50
x2 = 49 × 25
x = 35 and x = -35
Here since the x is a number between 1 to 49,
x = 35.

Hence, The value of x is 35.

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