Question

The houses of a row are number consecutively from 1 to 49. Show that there is a value of *x* such that the sum of numbers of the houses preceding the house numbered *x* is equal to the sum of the number of houses following it.

Find this value of *x*.

[Hint *S*_{x}_{ − 1 }= *S*_{49} − *S*_{x}]

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Solution

The number of houses was

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 *x*^{th }house was like this.

We know that,

Sum of *n* terms in an A.P.

Sum of number of houses preceding *x*^{th} house =* S*_{x}_{ − 1}

Sum of number of houses following *x*^{th} house = *S*_{49} − *S*_{x}

It is given that these sums are equal to each other.

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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