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If an A.P. if mth term is n and the nth term is m, where m ≠ n, find the pth term

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We have 

am = a + (m – 1)d = n, …… (1) 

and an = a + (n – 1)d = m …….. (2) 

Solving (1) and (2), we get 

(m, n) d = n – m, or d = -1, 

and a = n + m – 1 ……. (3) 

Therefore ap = a + (p – 1)d ……. (4) 

= n + m – 1 + (p – 1) (-1)

= n + m – p 

Hence, the pth term is n + m – p.

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