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If the pth term of an A.P. is q and the qth term is p, prove that its nth term is (p + q – n) - Mathematics

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Sum

If the pth term of an A.P. is q and the qth term is p, prove that its nth term is (p + q – n)

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Solution

Let a be the first term and d be the common difference of the given A.P. Then,

pth term = q ⇒ a + (p – 1) d = q ….(i)

qth term = p ⇒ a + (q – 1) d = p ….(ii)

Subtracting equation (ii) from equation (i),

we get

(p – q) d = (q – p) ⇒ d = – 1

Putting d = – 1 in equation (i), we get

a = (p + q – 1)

nth term = a + (n – 1) d

= (p + q – 1) + (n – 1) × (–1) = (p + q – n)

Concept: Arithmetic Progression
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