Introduction:
An arithmetic progression (A.P.) is a sequence of numbers in which the difference between any two consecutive terms is constant. In this case, we are given that the Nth term of the A.P. is a linear expression in n. We need to prove that the common difference of the A.P. is equal to the coefficient of n.

Proof:
To prove that the common difference of the A.P. is equal to the coefficient of n, we will consider the general formula for the Nth term of an A.P. and express it as a linear expression in n.

1. General Formula for the Nth term of an A.P.:
The general formula for the Nth term of an A.P. is given by:
an = a + (n-1)d

Where:
an = Nth term of the A.P.
a = First term of the A.P.
d = Common difference of the A.P.
n = Position of the term in the A.P.

2. Expressing the Nth term as a Linear Expression in n:
Let's express the Nth term (an) as a linear expression in n:

an = a + (n-1)d
= a + dn - d

Now, we can see that the expression contains two terms involving n:
1. dn (the coefficient of n is d)
2. -d (constant term)

3. Comparison with a Linear Expression:
To prove that an is a linear expression in n, we need to compare it with the general form of a linear expression:

an = mn + c

Where:
m = coefficient of n
c = constant term

Comparing the two expressions, we can observe that the coefficient of n in the linear expression mn + c is m, whereas in an, it is d.

4. Conclusion:
Since the Nth term of the A.P. can be expressed as a linear expression in n, and the coefficient of n in the linear expression is d (the common difference of the A.P.), we can conclude that the common difference of the A.P. is equal to the coefficient of n.

Therefore, if the Nth term of an A.P. is a linear expression in n, the common difference of the A.P. is indeed equal to the coefficient of n.