The houses of a row in a colony 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 houses following it . Find the value of x.
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House numbers: 1 , 2, 3, 4, .....x-1, x , x+1, x+2, ..... 48, 49
Sum 1 : 1+2+3+...+x-1 = (x-1) x /2
Sum 2: x+1, x+2, ..... 48+ 49
This is an AP. Number of terms: 49-x-1+1 = 49-x
Sum2 = (49-x) (x+1 + 49)/2
Sum1 = Sum2 => x(x-1) = (49-x)(x+50)
2x² = 49 * 50
x = 35
Sum 1 : 1+2+3+...+x-1 = (x-1) x /2
Sum 2: x+1, x+2, ..... 48+ 49
This is an AP. Number of terms: 49-x-1+1 = 49-x
Sum2 = (49-x) (x+1 + 49)/2
Sum1 = Sum2 => x(x-1) = (49-x)(x+50)
2x² = 49 * 50
x = 35
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