Introduction:
This question is related to Arithmetic Progression (AP) and requires finding the rth term of the given AP series.

Given:
The mth term of the AP is n and the nth term is m.

Method:
To find the rth term of the given AP series, we need to use the formula for the nth term of an AP and then replace the values of m and n in it.

The formula for the nth term of an AP is given as: a + (n-1)d
where, a is the first term of the AP, d is the common difference, and n is the term number.

Solution:
Let us assume that the first term of the AP is a and the common difference is d.

We are given that the mth term of the AP is n. So, from the formula, we can write:
n = a + (m-1)d ... (1)

Similarly, we are given that the nth term of the AP is m. So, from the formula, we can write:
m = a + (n-1)d ... (2)

Now, we can solve these two equations for a and d.

Subtracting equation (1) from equation (2), we get:
m - n = (n - m)d
or, d = (m - n) / (n - m) = -1

Substituting the value of d in equation (1), we get:
n = a + (m-1)(-1)
or, a = n + m - 1

Now, we can use the formula for the rth term of the AP:
ar = a + (r-1)d

Substituting the values of a and d, we get:
ar = (n + m - 1) - (r - 1)
or, ar = n + m - r

Therefore, the rth term of the given AP series is (n + m - r).

Conclusion:
In this way, we have found the rth term of the given AP series using the given information about its mth and nth terms.