Question

The houses in a row are numbered from 1 to 49.show that there exists a value of x such that the sum of numbers of houses preceeding the house numbered x is equal to sum of numbers of the houses following x...

Answer 1

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Answer 2

Answer:

x=35

Step-by-step explanation:

The numbers of houses preceding the house numbered $x=x-1$

Therefore sum of house preceding house number $x=S_{x-1}$

Sum of the number of houses following $x=S_{49}-S_{x}$

Since,

$S_{x-1}=S_{49}-S_{x}$...................equation-1

We know that sum of first $n$ natural number =$S_{n}=\frac{n(n+1)}{2}$

Therefore,

$S_{x}=\frac{(x)(x+1)}{2}$

$S_{x-1}=\frac{(x-1)(x-1+1)}{2}=\frac{(x-1)(x)}{2}$

$S_{49}=\frac{(49)(49+1)}{2}=\frac{(49\times50)}{2}$

Therefore

$S_{x-1}=S_{49}-S_{x}\\\\\Rightarrow\frac{(x-1)(x)}{2}=\frac{(49\times50)}{2}-\frac{(x)(x+1)}{2}\\\\\Rightarrow\frac{1}{2}(x-1)(x)=\frac{1}{2}[(49\times50)-(x)(x+1)]\\\\\Rightarrow{x}^{2}-x=[2450-x^{2}-x)]\\\\\Rightarrow{x}^{2}=2450-x^{2}\\\\\Rightarrow2{x}^{2}=2450\\\\\Rightarrow{x}^{2}=\frac{2450}{2}\\\\\Rightarrow{x}^{2}=1225\\\\\Rightarrow{x}=\sqrt{1225}\\\\\Rightarrow{x}=\pm35$

Since the number of house can not be negative so consider the value of x is 35.