Show that \[{(a - b)^2},{a^2} + {b^2}\] and \[{(a + b)^2}\] are in A.P.

Hint: We are given three terms and we need to find out if the given three terms are in arithmetic progression (AP) or not. Thus, for that, we need to find the difference between the two adjacent terms and check if we are getting the constant difference. Also, we can use the formula to find the third term and check if it is the same as what we are given in the question.

Complete step-by-step answer:
Here, we are given three terms and we need to show that they are in A.P.
Given that,
First term: \[{a_1} = {(a - b)^2}\]
Second term: \[{a_2} = {a^2} + {b^2}\]
Third term: \[{a_3} = {(a + b)^2}\]
We know that, in AP, the difference between two consecutive terms will be the same (or constant).
Thus, to prove that the given three terms are in AP, we will find the common difference between the two adjacent terms.
First, we will find the common difference between first and second term as below:
 \[{d_1} = {a_2} - {a_1}\]
Substituting the values in the above equation, we will get,
 \[ = {a^2} + {b^2} - [{(a - b)^2}] \]
We know that, \[{(a - b)^2} = {a^2} + {b^2} - 2ab\] and applying this, we will get,
 \[ = {a^2} + {b^2} - [{a^2} + {b^2} - 2ab] \]
Removing the brackets, we will get,
 \[
   = {a^2} + {b^2} - {a^2} - {b^2} + 2ab \\
   = 2ab \;
 \]
Second, we will find the difference between the second and third term as below:
 \[{d_2} = {a_3} - {a_2}\]
Substituting the values in the above equation, we will get,
 \[ = {(a + b)^2} - [{a^2} + {b^2}] \]
We know that, \[{(a + b)^2} = {a^2} + {b^2} + 2ab\] and applying this, we will get,
 \[ = {a^2} + {b^2} + 2ab - [{a^2} + {b^2}] \]
Removing the brackets, we will get,
 \[
   = {a^2} + {b^2} + 2ab - {a^2} - {b^2} \\
   = 2ab \;
 \]
Thus, we are getting the common difference which is the same.
Hence, the given three terms are in AP.

Another Method:
First we need to find the difference between first term and the second term
i.e. \[d = 2ab\]
Next we will use the formula to find the third term and check if the given third term is same as what we are getting, as below:
 \[\therefore {a_3} = {a_1} + 2d\]
 \[ \Rightarrow {a_3} = {(a - b)^2} + 2(2ab)\]
 \[ \Rightarrow {a_3} = {a^2} + {b^2} - 2ab + 4ab\]
 \[ \Rightarrow {a_3} = {a^2} + {b^2} + 2ab\]
 \[ \Rightarrow {a_3} = {(a + b)^2}\]
Hence, the third term is the same as what is given in the question. So, all the terms are in AP.

Note: An Arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. There are three different types of progressions and they are arithmetic progression (AP), geometric progression (GP) and harmonic progression (HP). The relation between the three is \[AP \geqslant GP \geqslant HP\] and \[G{P^2} = AP \times HP\] .