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If m times the mth term …

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If m times the mth term of an AP is equal to the nth term show that (m+n)th term of that AP is zero
  • 1 answers

Sia ? 5 years, 5 months ago

Let a be the first term and d be the common difference of the given AP. Then, in general the mth and nth terms can be written as 
Tm = a + (m - 1) d and Tn = a + (n - 1)d respectively.
According to the question,we are given that,
 (m.Tm) = (n.Tn)
{tex}\Rightarrow{/tex} m.{a + (m - 1)d} = n.{a + (n - 1)d}
{tex}\Rightarrow{/tex} a.(m - n) + {(m2 - n2) - (m - n)} . d = 0
{tex}\Rightarrow{/tex} (m - n).{a + (m + n - 1)}d.
{tex}\Rightarrow{/tex} (m - n).Tm+n = 0
{tex}\Rightarrow{/tex}Tm+n = 0 [{tex}\because{/tex} (m-n){tex}\neq{/tex}0].
Hence, the (m + n)th term is zero.

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