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Question

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

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Solution

Let be first term =a

common difference =d

Tm=a+(m1)d....(i)

Tn=a+(n1)d....(ii)

According to question,
mTm=nTn

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

ma+m(m1)d=na+n(n1)d

mana+m(m1)dn(n1)d=0

a(mn)+d[m2mn2+n]=0

Put the value of a in equation (iii)

Tm+n=d(m+n1)+(m+n1)d

Tm+n=0

(m+n)th term of A.P. is zero.

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