If theta = frac{pi}{6},then the 10^{th} term | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The given series is a geometric progression where the first term $a = 1$ and the common ratio $r = (\cos 	heta + i \sin 	heta) = e^{i 	heta}$.T

Vedclass · Accepted Answer

$-i$

Vedclass · Answer

(D) The given series is a geometric progression where the first term $a = 1$ and the common ratio $r = (\cos 	heta + i \sin 	heta) = e^{i 	heta}$.The $n^{th}$ term of a geometric progression is given by $T_n = a \cdot r^{n-1}$.For the $10^{th}$ term,$n = 10$,so $T_{10} = 1 \cdot (e^{i 	heta})^{10-1} = e^{i 9 	heta}$.Given $	heta = \frac{\pi}{6}$,we substitute the value:$T_{10} = e^{i 9 (\frac{\pi}{6})} = e^{i \frac{3\pi}{2}}$.Using Euler's formula $e^{i \phi} = \cos \phi + i \sin \phi$:$T_{10} = \cos(\frac{3\pi}{2}) + i \sin(\frac{3\pi}{2}) = 0 + i(-1) = -i$.

If $\theta = \frac{\pi}{6}$,then the $10^{th}$ term of the series $1 + (\cos \theta + i \sin \theta) + (\cos \theta + i \sin \theta)^2 + (\cos \theta + i \sin \theta)^3 + \dots$ is equal to:

Similar Questions

The atomic numbers of elements $A$,$B$,$C$ and $D$ are $Z-1$,$Z$,$Z+1$ and $Z+2$,respectively. If $B$ is a noble gas,choose the correct statements from the following:
$(1)$ $A$ has high electron affinity
$(2)$ $C$ exists in $+2$ oxidation state
$(3)$ $D$ is an alkaline earth metal

$C-C$ bond length is maximum in

Which one of the following elements in plants is not remobilised?

In an emission spectrum,the transitions $n_2 \to n_1, n_3 \to n_1, n_4 \to n_1, n_5 \to n_1$ result in the emission of energies $E_1, E_2, E_3, E_4$ respectively. What is the increasing order of these energies?

The lines represented by the equation $x^2-y^2-x+3y-2=0$ are :

Vedclass Test Series

Exam Paper Generator

Online Exam Module