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Question
The houses in a row are numbered consecutively from 1 to 49. Show that there is a value m such that the sum of numbers of the houses preceding the house marked m is equal to the sum of the numbers of houses following it. Find the value 'm'.
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Solution
1−−−n−1−−−49
Sum from 1 to m−1=Sum from m+1 to 49
AP with a=1, an=m−1 AP with a=m+1, an=49
n=m−1 n=49−(m+1)+1
=49−m−1+1
= 49 -m
m−12(1+m−1)=49−m2(m+1+49)
m−12(m)=49−m2(m+50)
m(m−1)=(49−m)(m+50)
m2−m=49m+2450−m2−50m
2m2−m=2450−m
2m2=2450⇒m2=1225⇒m=√1225=35
Sum from 1 to m−1=Sum from m+1 to 49
AP with a=1, an=m−1 AP with a=m+1, an=49
n=m−1 n=49−(m+1)+1
=49−m−1+1
= 49 -m
m−12(1+m−1)=49−m2(m+1+49)
m−12(m)=49−m2(m+50)
m(m−1)=(49−m)(m+50)
m2−m=49m+2450−m2−50m
2m2−m=2450−m
2m2=2450⇒m2=1225⇒m=√1225=35
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