G285 2021 Fall Problem Set Problem 8
Problem
If the value of can be represented as
, where
and
are relatively prime. Find
.
Solution
We begin with a simpler problem . Now, suppose
and
are constant. We have a converging geometric series for
with a sum of
. Now, make
everchanging. We have $\frac{1}{4^{b+c}=\left(\frac{4}{3} \right)^2 = \frac{16}{9}$ (Error compiling LaTeX. Unknown error_msg), so the entire sum must be
.
Now, coming back to the original problem, we split the single sum into :