High School

The third, fifth, and eighth terms of an arithmetic progression (AP) are the first three consecutive terms of a geometric progression (GP). Given that the first term of the AP is 8, calculate the common difference of the AP.

Answer :

Answer:

The common difference = 2.

Step-by-step explanation:

An AP can be written as a1, a1 + d, a1 + 2d, a1 + 3d, a1 + 4d, a1 + 5d, a1 + 6d , a1 + 7d.

where a1 = first term and d is the common difference.

Here first term = a1 = 8

3rd term = a1 + 2d = 8 + 2d

5th term = a1 + 4d = 8 + 4d

8th term = 8 + 7d

First 3 terms of a GP are a , ar and ar^2

So from the given information:

a = 8 + 2d

ar = 8 + 4d

ar^2= 8 + 7d

Dividing the second equation by the first we have

r = (8 + 4d)/(8 + 2d)

Dividing the third by the second:

r = (8 + 7d) / (8 + 4d)

Therefore, eliminating r we have:

(8 + 4d)/(8 + 2d) = (8 + 7d)/(8 + 4d)

(8 + 4d)^2 = (8 + 2d)(8 + 7d)

64 + 64d + 16d^2 = 64 + 72d^ + 14d^2

2d^2 - 8d = 0

2d(d^2 - 4) = 0

2d = 0 or d^2 = 4, so

d = 0, 2.

The common difference can't be zero so it must be 2.

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