9758/2022/P1/Q09

9 (a) An arithmetic series has first term $a$ and common difference $d$, where $d \neq 0$. The first, third and fifteenth terms of this series are the first, second and third terms of a geometric series. Find the common difference d in terms of $a$.

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(b) A geometric series has first term $\sin \theta$ and common ratio $-\cos \theta$, where $0<\theta<\frac{\pi}{2}$.
(i) Show that the sum to infinity of this series is $\tan k \theta$, where $k$ is a constant to be found.

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(ii) Given that $\theta=\frac{\pi}{3}$, find the exact sum of the first seven terms of this series.

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