Final answer:
The resulting terms after each term in a GP is increased by the middle term will be in an Arithmetic Progression (AP), as the differences between the new terms are constant.
Step-by-step explanation:
If three consecutive terms of a Geometric Progression (GP) are increased by their middle term, the resulting terms will be in Arithmetic Progression (AP). To illustrate this, let's denote the three terms of the GP as a/r, a, and ar where a is the middle term and r is the common ratio of the GP. When we add the middle term a to each, the resulting terms are a/r + a, a + a, and ar + a which simplifies to a(1 + 1/r), 2a, and a(r + 1). The difference between the consecutive terms is a(1 - 1/r) and a(r - 1), and since these differences are constant, the resulting terms are in AP.
For example, let's say the three consecutive terms of the GP are a, ar, and ar^2 (where r is the common ratio). When you increase each term by the middle term (ar), you get a+ar, ar+ar^2, and ar^2+ar^3. These terms are still in a GP, with the same common ratio r.
Therefore, the correct answer is c) They will still be in GP.