The first 3 terms of an arithmetic progression are \(2\tan2x,\ 5\tan2x,\ 8\tan2x\). Find the values of \(x\), where \(-180^\circ\leq x\leq180^\circ\), for which the sum to 30 terms is \(455\sqrt3\).[5]
The first 3 terms of a geometric progression are \(5\cos^2\left(\theta-\dfrac\pi2\right),\ 20\cos^4\left(\theta-\dfrac\pi2\right),\ 80\cos^6\left(\theta-\dfrac\pi2\right)\), where \(-\dfrac\pi6\leq\theta\leq\dfrac{7\pi}6\). Find the values of \(\theta\) for which this geometric progression has a sum to infinity.[6]
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Answer:(a) \(-165^\circ,-75^\circ,15^\circ,105^\circ\). (b) Official answer: \(-\dfrac\pi6<\theta<\dfrac\pi6\), excluding \(0\), or \(\dfrac{5\pi}6<\theta<\dfrac{7\pi}6\), excluding \(\pi\).