If A, B and C are three arithmetic progressions (APs) as given below :

A = 2, 4, 6, 8, …….. upto n terms

B = 3, 6, 9, 12, …… upto n terms

C = 0, 4, 8, 12, …… upto n terms, then

out of A + B, A - C, C - B and B - A which is/are A.P. ?

1. A + B

2. A - C

3. C - B

4. All are A.P.

Given,

A = 2, 4, 6, 8, …….. upto n terms

B = 3, 6, 9, 12, …… upto n terms

C = 0, 4, 8, 12, …… upto n terms, then

A + B = 2 + 3, 4 + 6, 6 + 9, 8 + 12, ……..

= 5, 10, 15, 20, ……….

The above list is an A.P. with common difference = 5.

A - C = 2 - 0, 4 - 4, 6 - 8, 8 - 12, ……

= 2, 0, -2, -4, ………

The above list is an A.P. with common difference = -2.

C - B = 0 - 3, 4 - 6, 8 - 9, 12 - 12,………

= -3, -2, -1, 0, ………

The above list is an A.P. with common difference = 1.

Hence, Option 4 is the correct option.