Math Help - Pennies

1. Pennies

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?

2. Originally Posted by RTC1996
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?
$\displaystyle a + ar + ar^2 + ar^3 + ... + ar^n = \frac{a(1 - r^{n+1})}{1-r}$

3. a,r and n

What do a, r and n stand for in your formula?

4. He showed that: the sum was $a+ ar+ ar^2+ \cdot\cdot\cdot + ar^n$ 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 $1+ 2+ 4+ \cdots+ 2^{19}+ 2^{20}$ and $1+ 2+ 4+ \cdot\cdot\cdot+ 2^{29}+ 2^{20}$. Compare that to $a+ ar+ ar^2+ \cdot\cdot\cdot+ ar^n$. What do you think a, r, and n are in those sums?

5. Originally Posted by RTC1996
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?
You're expected to spot a pattern.

Day 1: 1
Day 2: 2x1 = 2.
Day 3: 2x2 = 4.
Day 4: 2x4 = 8.
Day 5: 2x8 = 16.

etc.

Spot the pattern, make a conjecture, state the answers.

6. 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.

7. Originally Posted by TheGreenLaser
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.

[snip]
No, look again. skeeter's formula is the usual 'sum of a geometric series' formula, correctly suggested for this question if the OP choses appropriate values for a and r.

8. 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?

9. Originally Posted by TheGreenLaser
Maybe I'm just misunderstanding the question... isn't he just looking for a specific term in the series, not a sum? [snip]
The question asks: "How much money will you have at the end of 20 days?"

To me this implies the total of all money received up to and including the 20th day. Your interpretation is "How much money will you get on the 20th day?