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Question

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

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Solution

Let a be the first term and d be the common difference of the given A.P.
Also, let tm be the (m)th term and tn be the (n)th term of the progression.
Given that,
m×tm=n×tnmtm=ntn
For an A.P., we know that, an=a+(n1)d
mtm=ntnm[a+(m1)d]=n[a+(n1)d]
m[a+(m1)d]n[a+(n1)d]=0
a(mn)+[m(m1)n(n1)]d=0
a(mn)+[(m2n2)(mn)]d=0
a(mn)+(mn)(m+n1)d=0
(mn)[a+(m+n1)d]=0
a+((m+n)1)d=0
am+n=0

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