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

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Sum

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

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Solution

Let a be the first term and d be the common difference of the given A.P. Then, m times mth term = n times nth term

⇒ mam = nan

⇒ m{a + (m – 1) d} = n {a + (n – 1) d}

⇒ m{a + (m – 1) d} – n{a + (n – 1) d} = 0

⇒ a(m – n) + {m (m – 1) – n(n – 1)} d = 0

⇒ a(m – n) + (m – n) (m + n – 1) d = 0

⇒ (m – n) {a + (m + n – 1) d} = 0

⇒ a + (m + n – 1) d = 0

⇒ am+n = 0

Hence, the (m + n)th term of the given A.P. is zero

Concept: Arithmetic Progression
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