4.02: Geometric progressions

In a geometric progression with first term \(a\ne0\) and common ratio \(r\), the fourth term is \(-8\) times the seventh term. Find \(r\).

The sum of the first five terms is \(\dfrac{99}{32}\). Find \(a\).

A different geometric progression has first three terms \(\cos\theta\sin\theta\), \(\cos\theta\sin^3\theta\) and \(\cos\theta\sin^5\theta\), where \(0<\theta<\dfrac{\pi}{2}\). Show that its sum to infinity is \(\tan\theta\).