(Q4).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 and find this value of x. [Hint: Sx-1 = S49 – Sx]

Given: Houses with numbers from 1 to 49.

x is a number x such that,

Sx-1 = S49 – Sx

We know that, Sx =
x
2
 [ 2a + (x - 1)d]

where a = ; d = a2 - a1 = - =

∴ Sx-1 =
x-1
2
 [2 × + (x - 1 - )× ]
= (
x-1
2
 ) [ + x - ] =
(x-1)()
2

Also,

S49 - Sx =
2
 [2 × + (49 - )× ] -
2
 [2 × + (x - )× ]
=
2
 [ + ] -
2
 [ + x - ]
=
×
2
 -
(x+)
2
= × -
(x+)
2
= -
(x+)
2

By problem,

x(x-1)
2
 = -
x(x+1)
2
⇒
x(x-1)
2
 +
x(x+1)
2
 =

⇒ x(x – 1) + x(x + 1) = ×

⇒ 2 – x + 2 + x =

⇒ x2 =

⇒ 2 =
=

⇒ x = √ = ±

∴ x = [∵ x is a counting number]