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Algebra Level 2

The houses in a row are numbered consecutively from 1 to 49.Show that there is 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 house following it. Find the value of 'x'.


The answer is 35.

Anandhu Raj by Anandhu Raj , 23, India

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2 solutions

Prasun Biswas
Feb 19, 2014

We should first know the identity 1 + 2 + 3 + . . . + x = x ( x + 1 ) 2 1+2+3+...+x=\frac{x(x+1)}{2} to solve this problem. According to the problem, we have ---

x ( x + 1 ) 2 = 49 × ( 49 + 1 ) 2 − ( x − 1 ) x 2 \frac{x(x+1)}{2}=\frac{49\times (49+1)}{2}- \frac{(x-1)x}{2} [since last house no. = 49]

x ( x + 1 ) 2 = 49 × 50 − x ( x − 1 ) 2 \frac{x(x+1)}{2}=\frac{49\times 50 - x(x-1)}{2}

x ( x + 1 ) = 2450 − x ( x − 1 ) x(x+1)=2450 - x(x-1)

x 2 + x = 2450 − x 2 + x ⟹ 2 x 2 = 2450 ⟹ x 2 = 1225 ⟹ x = ± 35 x^2+x=2450-x^2+x \implies 2x^2=2450 \implies x^2=1225 \implies x=\boxed{\pm 35}

But, house number x x cannot be (-ve), so our required value of x = 35 x=\boxed{35}

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Leo Prakash
Mar 4, 2014

sum(x-1) = sum(49) - sum(x)

i.e sum(x) = x*(x+1)/2

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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