NP problem?

Algebra Level 2

In an arithmetic progression if m th m^\text{th} term is equals to n n while the n th n^\text{th} term is equals to m m , with m ≠ n m\ne n . Find the p th p^\text{th} term.

m-n+p n+m-p n+m+p n-m+p

Akshat Sharda by Akshat Sharda , 21, India

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1 solution

Tom Engelsman
May 15, 2020

Let:

a m = a 1 + ( m − 1 ) δ = n a_{m} = a_{1} + (m-1)\delta = n (i)

a n = a 1 + ( n − 1 ) δ = m a_{n} = a_{1} + (n-1)\delta = m (ii)

Subtracting (i) from (ii) produces: − δ ( m − n ) = m − n ⇒ δ = − 1 . -\delta (m-n) = m - n \Rightarrow \boxed{\delta = -1}. Substituting this common-difference value into either (i) or (ii) yields a 1 = m + n − 1 . \boxed{a_{1} = m+ n-1}. These two combined values finally give the p-th term as: a p = a 1 + ( p − 1 ) δ = ( m + n − 1 ) + ( p − 1 ) ( − 1 ) = m + n − p . a_{p} = a_{1} + (p-1)\delta = (m+n-1) + (p-1)(-1) = \boxed{m+n-p}.

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