Question

In an A.P. if m th term is n and the n th term is m , where m≠n, find the p th term​

Answer 1

Answer:

let the mth term  $A_{m} =n$ ⇒ $A_{m}= a+(m-1)d$ be eqn 1 and $A_{n}=m$ ⇒

$A_{n}= a+(n-1)d$ be eqn 2

subtracting eqn 2-eqn 1 ⇒ (2-1)⇒

$d(n-1)-d(m-1) = m-n$ ⇒

$dn-d-dm+d=m-n$ ⇒

$dn-dm=m-n$ ⇒

$d(n-m)= m-n$ ⇒

$d(\frac{n-m}{m-n}) = \frac{m-n}{n-m}$ ⇒

∴$d=1$

substituting d=1 in eqn (1) ⇒

$n= a+(m-1)1$ ⇒

$n=a+m-1$ ⇒

∴$n-m+1=a$

The pth term therefore is

$A_{p}=a+(p-1)d$ ⇒

$n-m+1+(p-1)1$ = $A_{p}$ ⇒

$n-m+1+p-1 = A_{p}$ ⇒

$n-m+p=A_{p}$ ⇒

∴The pth term

$A_{p}=n-m+p$

Step-by-step explanation: