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 $a$.

[4]

(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.

[3]

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

[2]

(a) Since they are consecutive terms of a geometric progression,
$$
\begin{array}{l}
\frac{a+14 d}{a+2 d}=\frac{a+2 d}{a} \\
a^2+14 a d=a^2+4 a d+4 d^2 \\
10 a d-4 d^2=0
\end{array}
$$
Since $d \neq 0$, then $10 a-4 d=0 \Rightarrow d=2.5 a$.
(b)
$$
\begin{array}{l}
\frac{\sin \theta}{1+\cos \theta} \\
=\frac{2 \sin \frac{1}{2} \theta \cos \frac{1}{2} \theta}{2 \cos ^2 \frac{1}{2} \theta} \\
=\tan \frac{1}{2} \theta
\end{array}
$$
Thus, $k=\frac{1}{2}$.
(c)
$$
\begin{array}{l}
\frac{\sin \frac{\pi}{3}\left[1-\left(-\cos \frac{\pi}{3}\right)^7\right]}{1+\cos \frac{\pi}{3}} \\
=\frac{0.5 \sqrt{3}\left[1-(-0.5)^{\top}\right]}{1+0.5} \\
=\frac{43 \sqrt{3}}{128}
\end{array}
$$