Solution:

To prove that the 11th term of an arithmetic progression (A.P.) cannot be of the form n^2 - 1, we need to consider the general form of an A.P. and analyze the pattern of its terms.

Arithmetic Progression:
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. The general form of an A.P. is given by:

a, a + d, a + 2d, a + 3d, ..., a + (n-1)d

where a is the first term, d is the common difference, and n is the number of terms.

Assumption:
Let's assume that the 11th term of the A.P. is of the form n^2 - 1.

Pattern Analysis:
To analyze the pattern of the terms, we can substitute the values of n = 1, 2, 3, ..., 11 in the expression n^2 - 1 and observe the resulting terms.

For n = 1, the term becomes 1^2 - 1 = 0
For n = 2, the term becomes 2^2 - 1 = 3
For n = 3, the term becomes 3^2 - 1 = 8
For n = 4, the term becomes 4^2 - 1 = 15
For n = 5, the term becomes 5^2 - 1 = 24
For n = 6, the term becomes 6^2 - 1 = 35
For n = 7, the term becomes 7^2 - 1 = 48
For n = 8, the term becomes 8^2 - 1 = 63
For n = 9, the term becomes 9^2 - 1 = 80
For n = 10, the term becomes 10^2 - 1 = 99
For n = 11, the term becomes 11^2 - 1 = 120

Observation:
From the above observations, we can see that the terms of the A.P. defined by the expression n^2 - 1 do not form a constant difference sequence. Therefore, the 11th term cannot be represented as n^2 - 1.

Conclusion:
Hence, we have proved that the 11th term of an arithmetic progression cannot be of the form n^2 - 1.