Hello, margaritas!

Here's the first part . . .

The first, second and fourth terms of a convergent geometric progression

are consecutive terms of an arithmetic progression.

Prove that the common ratio of the geometric progression is

The first, second and fourth terms of a GP are:

Three consecutive terms of an AP are:

[Yes, I let be the first term of both progressions.]

Then we have: .

Divide (2) by (1): .

And we have: .

The Quadratic Formula gives us: .

Therefore, the ratio is: .

Note: We are told that the geometric series is convergent.

. . . . .Hence:

. . . . .We must discard the other quadratic root.