Arithmetic Mean

To solve the problem, we need to analyze the progression and conditions given:

1. The sequence is a Geometric Progression (GP) with the first term and the common ratio , so .
2. Each term is the arithmetic mean of the next two terms. This implies:

1. For a GP, the terms can be expressed as:

1. Using the condition :

1. Factorize the equation:

Since , we solve . Using the quadratic formula:

1. This gives two possible values for : or .
2. For a valid GP that diverges as described, use because the other root will result in a negative sequence.
3. Now calculate . Since and using the sum formula for a GP:

1. Thus, simplifies to:

1. Substitute and simplify using the properties of the roots for this special sequence:

1. Therefore, .

The answer is, therefore, , which is option .