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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 preceding the house numbered X is equal to sum of the numbers of houses following X.


Solution

Formula,

$$S_n=\dfrac{n(n+1)}{2}$$

Given,

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

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

$$x(x-1)=49(50)-x(x+1)$$

$$x^2-x=2450-x^2-x$$

$$2x^2=2450$$

$$x^2=1225$$

$$\therefore x=35$$

Therefore number of houses is 35.

Mathematics

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