The eight,fourth and second terms of an arithmetic progression form the first three terms of a geometric series. The arithmetic progression has first term A and common difference d, and the geometric progression has first term G and common ratio r.

a)Given that d is not equal to 0,find the value of r
b)Given that A=2,find the sum to infinity of the geometric progression

8th term of AS = a+7d

4th term of AS = a+3d
2nd term of AS = a+d

then, if they form the first three terms of a GS
(a+3d)/(a+7d) = (a+d)/(a+3d)
(a+3d)^2 = (a+7d)(a+d)
a^2 + 6ad + 9d^2 = a^2 + 8ad + 7d^2
2d^2 - 2ad = 0
d(d-a) = 0
d = 0 or d = a
but we are told that d ≠ 0 , so
d = a

r = (a+3d)/(a+7d)
= 4a/(8a) = 1/2

f) if a = 2
sum(all terms) = a/(1-r)
= 2/(1 - 1/2)
= 4

To find the value of r in the geometric progression, we need to first find the eight, fourth, and second terms of the arithmetic progression.

Given:
First term of the arithmetic progression (A) = ?
Common difference of the arithmetic progression (d) = ?
First term of the geometric progression (G) = ?
Common ratio of the geometric progression (r) = ?

Let's start by finding the eight, fourth, and second terms of the arithmetic progression.

The general formula for the nth term of an arithmetic progression is:
An = A + (n-1)d

Using this formula, we can find the eight term (A8), fourth term (A4), and second term (A2) as:
A8 = A + 7d
A4 = A + 3d
A2 = A + d

Now, it is given that A2, A4, and A8 form a geometric progression.

In a geometric progression, the ratio of any two consecutive terms is constant.

So, we can write the following equation:
A4/A2 = A8/A4

Substituting the values of A4, A2, and A8, we get:
(A + 3d)/(A + d) = (A + 7d)/(A + 3d)

To find the value of r, let's simplify this equation.

Cross-multiplying, we get:
(A + 3d)(A + 3d) = (A + d)(A + 7d)

Expanding and simplifying, we get:
A^2 + 6Ad + 9d^2 = A^2 + 8Ad + 7d^2

Canceling out the A^2 terms, we get:
6Ad + 9d^2 = 8Ad + 7d^2

Rearranging the terms, we get:
6Ad - 8Ad = 7d^2 - 9d^2

Simplifying further, we get:
-2Ad = -2d^2

Cancelling out the common factor of -2d, we get:
A = d

Since it is given that d is not equal to 0, we can conclude that A is also not equal to 0.

Now, let's move on to part b) and find the sum to infinity of the geometric progression.

We know that the first term of the geometric progression (G) is 2.

The formula to find the sum to infinity of a geometric progression is:
Sum to Infinity (S) = G / (1 - r)

Substituting the value of G (2) into the formula, we get:
S = 2 / (1 - r)

Therefore, the sum to infinity of the geometric progression is 2 / (1 - r).