3.3: Trigonometric AP sum and convergent GP

The first three terms of an arithmetic progression are $\sin 3x$, $5\sin 3x$, $9\sin 3x$. Find the exact values of $x$, where $0\leqslant x\leqslant\dfrac{\pi}{2}$, for which the sum to twenty terms is equal to $390$.

The first three terms of a geometric progression are $20\cos y$, $10\cos^{2} y$, $5\cos^{3} y$.

Explain why this progression has a sum to infinity.

Find the value of $y$, where $y$ is in radians and $0<y<\dfrac{\pi}{2}$, for which the sum to infinity is $9$. Give your answer correct to $2$ decimal places.