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I've been a math and economics tutor here at UT Austin for the past 5 years, and I'm pretty sure that most students wouldn't have been able to follow your derivation. Here are the two places where I believe students would have problems:
1) When you started using your Savings term, you didn't explain where its various terms came from, for example, the p*1 term is the last payment that didn't have time to earn any interest, while the p*a^(t-1) term is the first payment that accrued interest for t-1 periods. Personally, I would have ordered the term in the reverse order that you did, so that the first term is from the first payment and the last term is from the last payment.
2) When you did the step where you multiplied by a and subtracted to cancel all the intermediate terms (one of my favorite slick-math-tricks, BTW), very few of my students would be able to follow it unless you included a couple of intermediate steps.
Maybe you're targeting your lectures only towards very advanced students, in which case, let me know and I'll stop the comments.