I'd like to get a good book on finite summations and products before I study infinite series more in depth next year. The book should cover geometric/ harmonic sums and prove different formulas for finite products. Some geometric intuition thrown in with the algebra/ induction would of course be a plus. Does anyone have a good recommendation? If no such book on just finite sums and products exists then do you know of a good book which covers them as part of a broader topic?
I'm not sure if it's exactly what you're looking for, but just about any intro to combinatorics (enumerative combinatorics) would contain a lot of these kinds of things. One book that comes to mind is "Discrete Mathematics and its Applications" by Rosen. It is a text for intro proof writing (so it'll contain induction and the like) and will certainly see a lot of finite sums involving binomial coefficients (and it may help you learn to count).