The pth term of an AP is q and qth term is p. Find its (p+q)th term.

Given p^th term of AP is q so using nth term formula
T_n =  a₁ + (n - 1)d ,                     
 a_p =  a₁+ d(p - 1)  = q ........(1)
Similarily q^th term of AP is p so ,

  a_q =  a₁ + d(q - 1) = p ........(2)
Now subtract these two equation we get , a_p -  a_q     

⇒  d (p - q) = (q - p)                           
     ⇒  d = -1 
Now put d = -1 in equation (1)  we get , 
  a₁ + d(q - 1) = p                         
a_{1 =} p + (q - 1) 

Now we have to find  a_(p+q)th term so  put the value of d and  a₁ we get ,   
      a_(p+q)th   =  a₁ + d(p + q - 1)           
  a_(p+q)th  =  p + q - 1 + d(p + q - 1)    = p + q -1 + (-1) (p + q - 1)  = 0

This gives (p + q)  term = 0