Question

Show that (a − b)², (a² + b²) and (a + b)² are in A.P.

Answer 1

Here, we are given three terms and we need to show that they are in A.P.,

First-term `(a_1) = (a - b)^2`

Second term `(a_2) = (a^2 + b^2)`

Third term `(a_3) = (a + b)^2`

So in an A.P, the difference of two adjacent terms is always constant. So to prove that terms are in A.P we find the common difference we get

`d = a_2 - a_1`

`d= (a^2 + b^2) - (a - b)^2`

`d = a^2 + b^2 - (a^2 + b^2 - 2ab)`

d = 2ab  ......(1)

Also

`d = a_3 - a_3`

`d = (a + b)^2 - (a^2 + b^2)`

`d = a^2 + b^2 + 2ab - a^2 - b^2`

d = 2ab .....(2)

Now since in equations (1) and (2) the value of d are equal we can say that these term are in A.P withn 2ab ast the commnon difference .

hence proved