Verify that each of the following is an $ \mathrm{AP} $, and then write its next three terms.
$ a+b,(a+1)+b,(a+1)+(b+1), \ldots $

Given:

Given sequence is \( a+b,(a+1)+b,(a+1)+(b+1), \ldots \)

To do:

We have to verify whether the given sequence is an AP and write its next three terms.

Solution: 

In the given sequence,

$a_1=a+b, a_2= (a+1)+b, a_3=(a+1)+(b+1)$

$a_2-a_1=(a+1)+b-a+b=1$

$a_3-a_2=(a+1)+(b+1)-[(a+1)+b]=a+b+2-a-1-b=1$

Therefore,

$a_2-a_1=a_3-a_2$

The given sequence is an AP.

$d=1$

$a_4=a_3+d=(a+1)+(b+1)+1=(a+2)+(b+1)$

$a_5=a_4+d=(a+2)+(b+1)+1=(a+2)+(b+2)$

$a_6=a_5+d=(a+2)+(b+2)+1=(a+3)+(b+2)$

The next three terms of the given sequence are $(a+2)+(b+1), (a+2)+(b+2)$ and $(a+3)+(b+2)$.