Q. 153.9( 8 Votes )

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

Answer :

__We know that nth term of A.P. , t _{n} = a + (n – 1) d.__

⇒ First, t_{n} = a + (n – 1) d

⇒ Then, t_{m} = a + (m – 1) d

Given, mt_{m} = nt_{n}

⇒ m [a + (m – 1) d] = n [a + (n – 1) d]

⇒ ma + m^{2}d – md = na + n^{2}d – nd

⇒ ma – na + m^{2}d – n^{2}d – md + nd = 0

⇒ a (m – n) + d (m^{2} – n^{2}) – d (m – n) = 0

__We know that a ^{2} – b^{2} = (a – b) (a + b)__

⇒ (m – n) [a + d (m + n) – d] = 0

⇒ [a + d (m + n) – d] = 0

⇒ a + (m + n – 1) d = 0

∴ (m + n)th term, t_{m + n} = 0

Hence proved.

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