TITLE: Whats better: 1 million dollars in a month or a penny(USD) doubled (and added) every day for 30 days?
QUESTION [5 upvotes]: THis is a question that I remember when I was in the 5th grade that tested our logical reasoning skills. And it is a simple choice knowing that the pennies doubling every day is an exponential function, but I'm trying to figure out how to create a formula for this solution without adding everything up by hand.
I know this has to be some form of a recurrence relation, but I cant figure out how to solve it. so far I have written down 
$F(n) = 2F(n-1)$ which wolframalpha solves for me as $c_1 2^{n-1}$ (I don't know what the $c_1$ is, so I assume its a constant).
and then when I plug in 
$\sum_{n = 1}^{30} 2^{n-1}$ it spits out $\$1,073,741,823$
Which is wrong because when I sum up all the days I get $10,737,418.23
I think the main problem is the recurrence relation I set up, but its been so long since my discrete math class, I've forgotten completely how to set that up.

REPLY [2 votes]: You can solve this problem without using Wolfram Alpha, recurrence relations, or anything complicated.
Consider the number of pennies you will get via each method.
A million dollars is 100,000,000 pennies (i.e. $10^8$).
How about a number of pennies which doubles each day? Well, imagine the number of pennies you get in binary. On day 1 you will get 1 (binary) penny. On day 2 you will get 10 (binary) (i.e. 3 pennies), on day three 100 pennies etc., until the 30th day where you will get 100...000 pennies, i.e. 1 with 29 zeroes - each day (numbered $d$) you are gaining a number of pennies being in binary 1 with $d-1$ zeroes.
Now imagine adding them up:
               1
            + 10
           + 100
             ...
+ 10000000...000 (1 with 29 zeroes)
----------------
= 11111111...111 (30 ones)

Now clearly, 11111...11 (30 ones) + 1 will be 1 followed by 30 zeroes, i.e. $2^{30}$. So the number of pennies will be $2^{30}-1$. No arithmetic needed so far.
Now, is this greater or lesser than $10^8$ (what you would have received had you received a million dollars)? Let's do an order of magnitude comparison. It's well known that $2^{10}$ is $1024$ (hence a kilobyte is about a thousand bytes etc.), and $1024$ is roughly $10^3$. So $2^{30}-1$ is roughly $10^{30/10*3}$ i.e. $10^9$. So, roughly speaking you'll be ten times better off taking an amount of pennies that doubles each day.