A geometric progression has common ratio \(r\) and first term \(a\), where \(a\ne0\).
The 4th term is −8 times the 7th term.
Find the value of \(r\).[2]
The sum of the first 5 terms of this progression is \(\dfrac{99}{32}\).
Find the value of \(a\).[2]
The first three terms of a different geometric progression are \(\cos\theta\sin\theta\), \(\cos\theta\sin^3\theta\) and \(\cos\theta\sin^5\theta\), for \(0<\theta<\dfrac{\pi}{2}\).
Show that the sum to infinity of this progression is \(\tan\theta\).[4]
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Answer:(a)(i) \(r=-\dfrac12\) (ii) \(a=\dfrac92\), using the mark scheme’s non-zero-ratio convention. (b) \(\tan\theta\) (shown).