solve for r

r^64-1 = 2.5x10^7 (r-1)

any ideas?

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.

Arggh!!!
Sorry, looks like we would have to use some method such as Newtons's Method or some kind of "equation solver" program.

To solve for r in the equation r^64 - 1 = 2.5x10^7(r - 1), you can follow these steps:

1. Expand the equation by distributing 2.5x10^7 to the terms inside the parentheses:
r^64 - 1 = 2.5x10^7r - 2.5x10^7

2. Rearrange the terms so that the equation is in standard polynomial form:
r^64 - 2.5x10^7r + 2.5x10^7 - 1 = 0

3. Combine like terms:
r^64 - 2.5x10^7r + 2.5x10^7 - 1 = 0
r^64 - 2.5x10^7r + 2.499999x10^7 = 0 (You can approximate 2.5x10^7 - 1 as 2.499999x10^7 for simplicity.)

Now, to solve this equation, you would typically need to use numerical methods or software specifically designed to find the roots of higher-degree polynomials, such as Newton's method or software like Wolfram Alpha. Given that the equation is a 64th-degree polynomial, it is not feasible to solve by hand.

However, if you need an approximate solution, you can make use of graphing calculators, online graphing tools, or software that can plot the graph of the equation. By observing the graph, you can estimate the value(s) of r where the equation is equal to zero.