Question

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

a + b, (a + 1) + b, (a + 1) + (b + 1), (a + 2) + (b + 1), (a + 2) + (b + 2),....

Answer 1

a + b, (a + 1) + b, (a + 1) + (b + 1), (a + 2) + (b + 1), (a + 2) + (b + 2),.... 

Here,

First term (a) = a + b

`a_1 = ( a + 1) + b`

`a_1 = ( a + 1) + (b +1)`

Now, for the given to sequence to be an A.P,

Common difference (d) `= a_1 - a = a_2 -a_1`

Here,

`a_1 -a = a + 1 + b - a- b`

= 1

Also,

`a_2 - a_1 = a+1+b+1-a-1-b`

= 1

Since  `a_1 - a = a_2 -a_1`

Hence, the given sequence is an A.P and its common difference is d = 1