The 3rd and 8th terms of a geometric progression are \(6\) and \(1458\) respectively. Find the common ratio and the first term of this progression.[4]
The first 3 terms of a second geometric progression are \(\cos\theta,\ 2\cos^2\theta,\ 4\cos^3\theta\), where \(-90^\circ<\theta<90^\circ\). Find the values of \(\theta\) for which this geometric progression has a sum to infinity.[4]
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Answer:(a) Common ratio \(3\), first term \(\dfrac23\). (b) \(-90^\circ<\theta<-60^\circ\) or \(60^\circ<\theta<90^\circ\).