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.