The second, fourth and eighth terms of A.P forms a G.P, and the sum of third and fifth term is 20. Find the first five terms of the progression?

Let the common difference of the arithmetic progression be d and the first term be a.

The second term = a + d
The fourth term = a + 3d
The eighth term = a + 7d

Since the second, fourth and eighth terms form a geometric progression, we have:

(a + d)(a + 7d) = (a + 3d)^2
Expanding both sides, we get:
a^2 + 7ad + ad + 7d^2 = a^2 + 6ad + 9d^2
a^2 + 8ad + 7d^2 = a^2 + 6ad + 9d^2
2ad = 2d^2
a = d

Now, we know that the sum of the third and fifth term is 20:
a + 2d + a + 4d = 20
2a + 6d = 20
2d + 6d = 20
8d = 20
d = 2.5

Therefore, the first term a = d = 2.5

The first five terms of the progression are:
2.5, 5, 7.5, 10, 12.5

Let the first term of the A.P be a and the common difference be d.

Then, the second term = a + d
Fourth term = a + 3d
Eighth term = a + 7d

Since the second, fourth and eighth terms form a G.P, we have:
(a + 3d)/(a + d) = (a + 7d)/(a + 3d)
Cross-multiplying, we get:
(a + 3d)^2 = (a + d)(a + 7d)
a^2 + 9d^2 + 6ad = a^2 + 8ad + 7d^2
d^2 - 2ad = 0
d(d - 2a) = 0
Therefore, d = 0 or d = 2a

If d = 0, then the terms are equal and the progression is not possible.

If d = 2a, then
Fifth term = a + 4d = a + 8a = 9a
Third term = a + 2d = a + 4a = 5a

Given that the sum of third and fifth term is 20, we have:
5a + 9a = 20
14a = 20
a = 20/14 = 10/7

Therefore, the first five terms of the progression are:
a = 10/7
a + d = 30/7
a + 3d = 50/7
a + 4d = 70/7
a + 8d = 150/7