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the houses of a row are numbered consecutively from 1-49 .Show that there is a value to x such that the sum of the house preceding the house no;x is equal to the sum of the numbers of the houses following it.find x
The number of houses are 1, 2, 3 … 49

It can be observed that the number of houses are in an A.P. having  a  as 1 and  d  also as 1.

Let us assume that the number of  xth house was like this.

We know that,
Sum of  n  terms in an A.P
Sum of number of houses preceding  xth house =  S x ? 1

Sum of number of houses following  xth house =  S49 ?  Sx
 Now,  Sum of the number of all houses before x = Sum of the number of all houses after x

Sx ? 1  =  S 49 ?  S x

Therefore,
 However, the house numbers are positive integers.

The value of  x  will be 35 only.

Therefore, house number 35 is such that the sum of the numbers of houses preceding the house numbered 35 is equal to the sum of the numbers of the houses following it.


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