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.