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Question

The houses in a row are numbered consecutively from 1 to 49. Show that there exists a value of X such that sum of numbers of houses proceeding the house numbered X is equal to sum of the numbers of houses following X.


Solution

Given, the houses in a row numbered consecutively from 1 to 49.

Now, sum of numbers preceeding the number X

= X(X1)2

And, sum of numbers following the number X 

= 49(50)2X(X1)2X

= 2450X2+X2X2

= 2450X2X2

According to the given condition,

Sum of no's precceding X = Sum of no's following X 

X(X1)2=2450X2X2

X2X=2450X2X

2X2=2450

X2=1225

X=35

Hence, at X=35, sum of no. of houses preceeding the house no. X is equal to sum of the no, of houses following X.

Mathematics

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