(a)[3]
The first two terms of an arithmetic progression are $1$ and $\cos^2 x$ respectively. Show that the total of the first ten terms can be written in the form $a - b\sin^2 x$, where the constants $a$ and $b$ are to be determined.
(b(i))[2]
The first two terms of a geometric progression are $1$ and $\frac{1}{3}\tan^2 \theta$ respectively, where $0 < \theta < \frac{1}{2}\pi$. Find the range of values of $\theta$ for which the progression converges.
(b(ii))[2]
Find the exact value of the sum to infinity when $\theta = \frac{1}{6}\pi$.