High School

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ 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.