If you are writing an application for a student government position, you do not necessarily need a topic sentence. An application doesn't always need to follow the rules of an essay. Instead, what you should do in your first paragraph is give an overview or summary of why you are applying and what your qualifications are.
The most important strategy in writing this sort of application is to focus on the needs of the position rather than on yourself. Since you haven't mentioned the actual position, it's difficult to give specific advice. If the role involves attending meetings, you might emphasize that, for example, you are really good at note taking and thus would be able to take minutes or give examples of how you have held similar roles in student clubs and had specific successes.
The key to a successful application is giving credible evidence of how your skills or experience would enable you to make a valuable contribution to the student government.