Hello, Espionage!

This is a very "busy" problem . . .

Steven takes an arithmetic sequence with common difference

and a geometric sequence with common ratio ,

and adds them together, term by term, to form a new sequence.

If the first three terms of the new sequence are, in order, 3, 8, and 15,

and and are both positive integers,

what is the sum of the possible values of ?

We are told: .

From [1]: .

Substitute into [2]: .

Substitute into [3]: .

Quadratic Formula: .

Substitute into [4]: .

The sum of the possible -values is: .