posted by Mikal .
solve for r
r^64-1 = 2.5x10^7 (r-1)
r-1 is a factor of r^64 - 1 so use synthetic division (notice the pattern) to divide both sides by r-1 to get
r^63 + r^62 + ... + r + 1 = 2.5(10)^7
the left side is a geometric series with
a = 1
common ratio = r
n = 64
so the sum of the left side = 1(r^64 - 1)/63
then (r^63 - 1)/63 = 2.5(10^7
r^63 = 63(2.5)(10)^7 + 1
taking the 63rd root I got
r = 1.3922
OOOPS, just looked at my solution again and
I'm WRONG!!! (must have had an early morning synapse collapse)
The sum of my series should have said
1(r^64 - 1)/(r-1)
then r^64 - 1 = 2.5(10)^7(r-1)
which brings me back to what we started with.
Sorry, looks like we would have to use some method such as Newtons's Method or some kind of "equation solver" program.