Arithmetic Progression Question: I have no numbers and I am not sure how to proceed

Question

Three numbers are consecutive terms of a geometric progression. If we add 2 to the second number, the new progression becomes arithmetic. If we add 9 to the third number, the progression becomes geometric. Find the original numbers.

Answer 1

To have an arithmetic progression, you need a common difference.

The second talks of adding $2$ to the second number, so $a,ar+2,ar^2$ is an arithmetic progression, so $ar^2-(ar+2)=(ar+2)-a$.

Then you get back to geometric by adding $9$ to the third, so $\frac {ar+2}a=\frac {ar^2+9}{ar+2}$.

There are your two equations.