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Sum

If m times m^{th} term of an A.P. is equal to n times its n^{th} 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 m^{th} term = n times n^{th} term

⇒ ma_{m} = na_{n}

⇒ 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

⇒ a_{m+n} = 0

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

Concept: Arithmetic Progression

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