If you have one penny the first day of the month and double your savings each day (1 penny first day, 2 pennies the second day, 4 pennies the third day etc.) How much money will you have at the end of 20 days? 30 days?
He showed that: the sum was $\displaystyle a+ ar+ ar^2+ \cdot\cdot\cdot + ar^n$ so r is the factor that is increasing in power in each term, a is the constant factor, and n is the number of terms in the sum.
Your sums were $\displaystyle 1+ 2+ 4+ \cdots+ 2^{19}+ 2^{20}$ and $\displaystyle 1+ 2+ 4+ \cdot\cdot\cdot+ 2^{29}+ 2^{20}$. Compare that to $\displaystyle a+ ar+ ar^2+ \cdot\cdot\cdot+ ar^n$. What do you think a, r, and n are in those sums?
Be careful though. The formula skeeter gave is for if you're adding twice what you had the previous day to the total. ie, you have 1 on day one, then you're given 2 more on day 2 (your total is 3), then you're given 4 more on day 3 (your total is 1 + 2 + 4 = 7), etc.
I think your question is just that your total is doubled every day, so on day 1 your total is 1, on day 2 your total is 2, on day 3, your total is 4, etc. This is much simpler to calculate. Look at the list of calculations mr fantastic posted and try to come up with a formula that gives you the total, based on the day (call it 'n' if you'd like). In this case there's a very simple formula.
Maybe I'm just misunderstanding the question... isn't he just looking for a specific term in the series, not a sum? Sorry if I'm just making this worse, but to me it looks like he just needs to find terms 20 and 30 of a geometric series.
The question says "you have one penny the first day of the month and double your savings each day", which looks more similar to a percent interest problem, where you just have to find the term n, in the series.
If I'm completely out to lunch just ignore me, I'll leave you guys to help, but I really think that's what the question's asking for. Maybe the OP can help clarify exactly what the question is asking for?
Yeah, I definitely see what you mean. It's the phrase "double your savings each day" that's getting me. It sounds to me like a compounding interest problem, with an interest rate of 100% per day, compounded daily. Like on day 1 you have a singular penny, and that's your 'savings'. Day 2, you 'double your savings', so you now have 2 pennies and those 2 pennies represent your savings. Then day 3, again you 'double your savings', meaning you now have 4 pennies, and those represent your savings as of day 3. If you're adding one penny to your savings the first day, 2 pennies the second day, 4 pennies the third day, etc, you're not really "doubl[ing] your savings every day" at all. You're adding to your savings twice what you added the previous day.
I guess we can leave it up to the OP to decide. I don't think there's much trouble with the math itself either way, so there's not really much point arguing. The OP probably has more context based on the other stuff he/she is doing to be able to figure out which the question is asking for.