If pth term of an A.P is q and qth term of the A.P. is p, find its rth term.

Sure, here is the step by step solution:

Step 1: The general form of an arithmetic progression (A.P) is a, a+d, a+2d, a+3d, ..., where 'a' is the first term and 'd' is the common difference.

Step 2: According to the problem, the pth term of the A.P is q and the qth term of the A.P is p.

Step 3: We can write these conditions as follows:

a + (p-1)d = q ----(equation 1)

a + (q-1)d = p ----(equation 2)

Step 4: Subtract equation 2 from equation 1, we get:

(p-q)d = q - p

Step 5: Simplify the above equation to find 'd':

d = (q - p) / (p - q)

Step 6: Substitute 'd' in equation 1 to find 'a':

a = q - (p-1)[(q - p) / (p - q)]

Step 7: Simplify the above equation to find 'a':

a = (pq - p^2 + q) / (p - q)

Step 8: Now, we have 'a' and 'd', we can find the rth term of the A.P:

rth term = a + (r-1)d

rth term = [(pq - p^2 + q) / (p - q)] + (r-1)[(q - p) / (p - q)]

Step 9: Simplify the above equation to find the rth term:

rth term = [(pq - p^2 + q + rq - rp - q + p) / (p - q)]

Step 10: Simplify further to get the final answer:

rth term = (pq - p^2 + rq - rp) / (p - q)

So, the rth term of the A.P is (pq - p^2 + rq - rp) / (p - q).