What is the common difference of an A.P. if the first term is 2 and C(a8,a2)=C(a8,a5 +(2)) ?

We'll re-write the terms of the arithmetical
progression:

a8 = a1 + 7d, where a1 is the first terms and
d is the common difference.

a8 = 2 +
7d

a2 = a1 + d

a2 = 2 +
d

a5 = a1 + 4d

a5 = 2 +
4d 

We'll add 2 both sides:

a5
+ 2 = 4 + 4d

Now we'll re-write the constraint from
enunciation:

C(2 + 7d , 2+d) = C(2 + 7d , 4 +
4d)

Instead of C(2 + 7d , 4 + 4d), we'll write the
complementary combination of C(2 + 7d , 2+d) = C(2 + 7d , 2 + 7d - 2 -
d).

We'll combine and eliminate like
terms:

C(2 + 7d , 2+d) = C(2 + 7d ,
6d).

So, the constraint from enunciation will
become:

C(2 + 7d , 6d) = C(2 + 7d , 4 +
4d)

Since the terms are equal, we'll
get:

6d = 4 + 4d

We'll
subtract 4d both sides:

6d - 4d =
4

2d = 4

d =
2

The common difference of the given
arithmetical progresison is d = 2.