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Question

If m times the mth term of an A.P. is equal to n times nth term, show that the (m+n)th term of the A.P. is zero.

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Solution

Given,

nth term of AP =tn=a+(n1)d

mth term of AP =tm=a+(m1)d

mtm=ntn

m[a+(m1)d]=n[a+(n1)d]

m[a+(m1)d]n[a+(n1)d]=0

a(mn)+d[(m+n)(mn)(mn)]=0

(mn)[a+d((m+n)1)]=0

a+[(m+n)1]d=0

But tm+n=a+[(m+n)1]d

tm+n=0


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