Question

if p th is q and q th is p prove that p th term is (p+q-n)?

Answer 1

Let first term be a and common difference be d.
pth term = a + (p-1)d = q -----(i)
qth term = a + (q-1)d = p -----(ii)
(ii) - (i) gives
(q - p)d = p - q or d = -1
(i) gives a = q - (p-1)d = q - (p-1)(-1) = q + p - 1
Hence,
nth term = a + (n-1)d = q + p -1 + (n-1)(-1) = q + p -1 + n +1 = p + q - n

Answer 2

according to the given problem,
a+(p-1)d = q                     ---- 1
and a+(q- 1)d = p             ----- 2 
from subtracting 2 from 1,
(p-1)d - (q-1)d = q - p
d {(p-1)-(q-1)} =  q-p
d{ p-1- q +1} = q - p
d(p - q)= ( q - p)
d= -1                    ----------3
putting d=-1 in equation 1
a + (p-1)-1=q
a-p+1=q
so, a= ( p +p-1)               -----4
now, we know that
tn= a +(n-1)d
by putting the value of a and d 

tn=(p+q-1) + (n-1) - 1
= p+q-1-n+1
=(p+q-n)

hence proved