Discrete Geometric Moment Generating Function

Hi everybody,

I have a question regarding how to proceed with the following problem. I am required to produce a moment generating function for the following problem. I would like to know if the operation I performed is valid for this question. *I never understood mgf's that well.*

Question:

A balanced coin is flipped repeatedly until a head appears. Let X denote the number of flips needed. Define Y = 2X-1.

__Find the moment generating function for Y.__

My reasoning:

This is a geometric distribution (discrete random variable).

I applied the definition of the m.g.f.

Note: I'm new to latex. Please consider the (2X-1) as raised to the power of e, along with the t.

Is this the valid first step for this problem, and others like it?

Thanks for your time!