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Arithmetic Progression

In an A.P., i...

Question

In an A.P., if $m^{t h}$ term is n and $n^{t h}$ term is m, show that its $r^{t h}$ term is $(m + n - r)$ .

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Solution

$T_{m} = n T_{n} = m S h o w T_{r} = (m + n - r) T_{m} = a + (m - 1) d n = a + (m - 1) d \to (i) T_{n} = a + (n - 1) d m = a + (n - 1) d \to (i i) o n s u b s t r a c t i n g (i) f r o m (i i) \Rightarrow (n - m) = (m - 1 - n + 1) d \Rightarrow (n - m) = (m - n) d \Rightarrow d = - 1, p u t d = - 1 e q u a t i o n (i) w e g e t n = a + (m - 1) \times - 1 \Rightarrow a = (m + n - 1) \Rightarrow T_{r} = a + (r - 1) d \Rightarrow T_{r} = (m + n - 1) + (r - 1) \times - 1 \Rightarrow T_{r} = m + n - 1 r + 1 ∴ T_{r} = m + n - r$

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