Question 83790
If you look at the relationship between the number of pennies on the squares you'll see that the number is growing exponentially. Going from 1 to 2, the number is doubled. Continuing from 2 to 4, again the number is doubled. So to get to any term you must double the previous value. So this problem involves powers and exponents. If you wanted to get to the 10th term, you would start at the first term and double each term to get to the tenth term. To get to the 10th term, you must multiply 2 by itself 10 times. So to get to the 64th term, you must multiply 2 by itself 64 times, see the pattern? To get the pattern down officially, the sequence is {{{2^n}}} where n is any term you pick. We start off at n=0 to get the first term of 1 and we move from there. Hope a little background on this helps, so here we go.



a)To find out how many pennies are on the 32th square, simply evaluate 2^31 (we go to the n-1 term since we started off at n=0). Well it comes out to 2,147,483,648 pennies.
If you want to verify, you can double 1 to 2, 2 to 4, etc until you get there.
Divide this figure by 100 to get the value in dollars.
{{{2147483648/100=21474836.48}}}
So he would pay $21,474,836.48 alone for the 32nd square



b)To find the sum of any geometric series, i.e. how to find 1+2+4+8+...2,147,483,648, you would use the sum of a geometric series formula.  If I have a series of n terms the sum is
{{{S=(a(1-r^(n)))/(1-r)}}} don't worry about a, we will ignore it, a=1 right now
and r is the factor to go from term to term, in this case r=2 (we're doubling everything). 
So {{{S=(1-2^32)/(1-2)}}}
{{{S=-4294967295/-1}}}
{{{S=4294967295}}} so this means that there are a total of 4,294,967,295 pennies if 32 squares were filled.
Divide this figure by 100 to get the value in dollars.
{{{4294967295/100=42949672.95}}}
So if there were only 32 squares, then he would pay $42,949,672.95


c)To find how many pennies would fill the entire board, use the same formula but with 64 squares
{{{S=(1-2^64)/(1-2)}}}
{{{S=(-18446744073709551616)/(-1)}}}
{{{S=(18446744073709551616)}}}
So there are 18,446,744,073,709,551,616 pennies needed to fill the entire board. Divide this figure by 100 to get the value in dollars. 
{{{18446744073709551616/100=184467440737095516.16}}}
So he would have to pay $184,467,440,737,095,516.16 to the salesman.