Problem of the Day #79

POTD February 25, 2024

\sum\limits_{n=0}^\infty\frac{\sin (nx)}{3^n}, if \sin x=1/3 and 0 \leq x\leq \frac{\pi}{2}, can be written as \frac{a+b \sqrt c}{d}, where c is a positive integer not divisible by the square of any prime and a,b,c are positive integers such that \gcd (a,b,c)=1. Find a+b+c+d.