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  • 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 horses proceeding the house numbered X is equal to sum of the numbers of houses following X   

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 horses proceeding the house numbered X is equal to sum of the numbers of houses following X 

 

 

 

 

 
 
 
 
 

Answers (1)

Sum of numbers of houses preceding x = sum of the no. of houses following x 

S _(x-1) = S _{49} - S _x\\\\

sum of first n natural numbers = S_n = \frac{n (n+1)}{2}\\\\ \frac{(x-1)(x-1+1)}{2}= \frac{49(49+1)}{2} - \frac{x (x+1)}{2}\\\\ \frac{(x-1)(x)}{2}= \frac{49(50)}{2} - \frac{x (x+1)}{2}\\\\ x (x-1) = 49 \times 50 - x ( x+1) \\\\ x^2 - x = 2450 - x^2 - x \\ 2 x^2 = 2450 \\\\ x ^2 = 2450/2 = 1225\\\\ x = \sqrt{1225}\\\\ x = \pm 35

since x can not be negative , hence x = 35 

Posted by

Ravindra Pindel

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