If m times my term of an AP is equal to nth term . Show that it's (m+n)th is 0.
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m tm=n TN
m{ a+(m-1)d}=n{a+(n-1)d}
ma + m^2d-md = na + n^2d -nd
ma-na + m^2d-n^2d-md+ nd==0
a(m-n)+ (m-n)(m+n)d-d(m-n)=0
Divide by m-n
a+{m+n-1}d=0
So a(m+n)=0
m{ a+(m-1)d}=n{a+(n-1)d}
ma + m^2d-md = na + n^2d -nd
ma-na + m^2d-n^2d-md+ nd==0
a(m-n)+ (m-n)(m+n)d-d(m-n)=0
Divide by m-n
a+{m+n-1}d=0
So a(m+n)=0
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