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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.

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