If θ=(π)/6 , then the 10th term of 1+(cos θ+i sin θ)+(cos θ+i sin θ)^2 +(cos θ+i sin θ)^3+… is equal to

Correct answer: (D) -i. T_10=(cos θ+i sin θ)^9 =e^i 9 θ=e^i 9 (π)/6=e^i 3 (π)/2=-i

Q139 / 200 · AP EAMCET 2001 · Mathematics

If $\theta=\frac{\pi}{6}$, then the 10th term of $$ \begin{aligned} & 1+(\cos \theta+i \sin \theta)+(\cos \theta+i \sin \theta)^2 \\ & +(\cos \theta+i \sin \theta)^3+\ldots \text { is equal to } \end{aligned} $$

$\begin{aligned} T_{10}=(\cos \theta+i & \sin \theta)^9 \\ & =e^{i 9 \theta}=e^{i 9 \frac{\pi}{6}}=e^{i 3} \frac{\pi}{2}=-i\end{aligned}$

Hence, the correct answer is option: 'D'.