The first 3 terms of an arithmetic progression are \(\log_x3,\ \log_x81,\ \log_x2187\). Find the sum to \(n\) terms, giving your answer in the form \(k\log_x3\), where \(k\) is in terms of \(n\).[3]
The first 3 terms of a geometric progression are \(1,\ 3\tan^2\theta,\ 9\tan^4\theta\), for \(0<\theta<\dfrac\pi2\). Find the values of \(\theta\) for which this geometric progression has a sum to infinity.[4]
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