3.18: Log AP, exponential GP term, trig GP convergence

An AP has first three terms $\ln q$, $\ln(q^{4})$, $\ln(q^{7})$. Given $S_{n}=\ln(q^{4845})$, find $n$.

A GP has first three terms $p^{3x}$, $p^{x}$, $p^{-x}$. Find the $n$th term in the form $p^{(a+bn)x}$.

A GP has first three terms $\dfrac{3}{4}\cos^{2}\theta$, $\dfrac{9}{16}\cos^{4}\theta$, $\dfrac{27}{64}\cos^{6}\theta$ for $0<\theta<\pi$. Find the set of values of $\theta$ for which a sum to infinity exists.