Question

The houses of a row are numbered consecutively from 1 to $49.$ Show that there is a valueof $x$ such that the sum of the numbers of the houses preceding the house numbered $x$ isequal to the sum of the numbers of the houses following it. Find this value of $x.$

• A. $32$
• B. $45$
• C. $35$
• D. $31$

Answer 1

The correct option is C $35$

Let there be a value of $x$ such that the sum of the numbers of the houses preceding the house numbered $x$ is equal to the sum of the numbers of the houses following it.

Since,

Sum of numbers of houses preceding $x=$ Sum of the number of houses following $x$

$S_{x-1}=S_{49}-S_x$

Sum of first $n$ natural number $=S_n=\frac{n(n+1)}{2}$

$\Rightarrow$ $\frac{(x-1)(x-1+1)}{2}=\frac{49(49+1)}{2}-\frac{x(x+1)}{2}$

$\Rightarrow$ $\frac{x(x-1)}{2}=\frac{49\times 50}{2}-\frac{x(x+1)}{2}$

$\Rightarrow$ $x(x-1)=2450-x(x+1)$

$\Rightarrow$ $x^2-x=2450-x^2-x$

$\Rightarrow$ $2x^2=2450$

$\Rightarrow$ $x^2=1225$

$\Rightarrow$ $x=\sqrt{5^2\times 7^2}$

$\Rightarrow$ $x=\sqrt{{35}^2}$

$\therefore$ $x=\pm 35$

Since number of houses can't be negative.

$\therefore$ $x=35$