```Question 69027
If you look at the relationship between the number of wheat grains 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 grains of wheat are on the 24th square, simply evaluate 2^23 (we go to the n-1 term since we started off at n=0). Well it comes out to 8,388,608 grains of wheat.
If you want to verify, you can double 2 to 4, 4 to 8, etc until you get there.

b)This one might be a little tricky, but I'll give it a shot. You could painstakingly add up 2,4,8...8,388,608 but that's probably not going to happen.
Also, I don't know if its just me, but this seems a little advanced for algebra. Now I could go into great detail of how to derive this formula, but I'm not. To find the sum of any geometric series, i.e. how to find 1+2+4+8+...8,388,608, there is a neat shortcut. If you're curious search "Sum of geometric series". Anyway, if I have a series of n terms the sum is
{{{S=(a(1-r^(n+1)))/(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=(1-2^24)/(1-2)}}}
{{{S=-16777215/-1}}}
{{{S=16777215}}} so this means that there are a total of 16,777,215 grains of wheat if 24 squares were filled.

c)To find how many grains 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 grains of wheat on the board```