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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 proceeding the house numbered X is equal to sum of the numbers of houses following X.
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 proceeding the house numbered X is equal to sum of the numbers of houses following X.
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Solution
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Given 1,2,3,4,5,....49 consecutive numbers
x is the number such that
sum of preceding numbers of x = sum of following numbers of x
sum of ( 1,2,3,....x-1) =
sum of [(x+1), (x+2) ,....48,49]
(x-1)/2[1+x-1] =
(49-x)/2[x+1+49]
(x-1)x=(49-x)(x+50)
x²-x=49x+2450-x²-50x
x²-x =2450-x²-x
x²+x²-x+x=2450
2x²=2450
x²=2450/2
x²=1225
x=√1225
x=35
required number is x= 35
sum of (1,2,3.....34) = sum of (36,37,....49)
Given 1,2,3,4,5,....49 consecutive numbers
x is the number such that
sum of preceding numbers of x = sum of following numbers of x
sum of ( 1,2,3,....x-1) =
sum of [(x+1), (x+2) ,....48,49]
(x-1)/2[1+x-1] =
(49-x)/2[x+1+49]
(x-1)x=(49-x)(x+50)
x²-x=49x+2450-x²-50x
x²-x =2450-x²-x
x²+x²-x+x=2450
2x²=2450
x²=2450/2
x²=1225
x=√1225
x=35
required number is x= 35
sum of (1,2,3.....34) = sum of (36,37,....49)
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