Binomial Theorem And Its Simple Applications Question 213
Question: The value of $ \sum\limits_{r=0}^{n}{^{n}C_{r}\sin (rx)} $ is equal to
Options:
A) $ 2^{n}\cdot {{\cos }^{n}}\frac{x}{2}\cdot \sin \frac{nx}{2} $
B) $ 2^{n}\cdot sin^{n}\frac{x}{2}\cdot \cos \frac{nx}{2} $
C) $ {2^{n+1}}\cdot {{\cos }^{n}}\frac{x}{2}\cdot \sin \frac{nx}{2} $
D) $ {2^{n+1}}\cdot sin^{n}\frac{x}{2}\cdot \cos \frac{nx}{2} $
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Answer:
Correct Answer: A
Solution:
- [a] $ \sum\limits_{r=0}^{n}{^{n}C_{r}\sin ,rx=Im( \sum\limits_{r=0}^{n}{^{n}C_{r}e^{irx}} )} $
$ =Im( \sum\limits_{r=0}^{n}{^{n}C_{r}{{( e^{ix} )}^{r}}} )=Im( {{( 1+e^{ix} )}^{n}} ) $
$ =Im{{(1+\cos ,x+i,\sin ,x)}^{n}} $
$ =Im,{{(2,cos^{2}\frac{x}{2}+2i,sin\frac{x}{2}.cos\frac{x}{2})}^{n}} $
$ =Im,{{( 2\cos \frac{x}{2}( \cos \frac{x}{2}+i\sin \frac{x}{2} ) )}^{n}} $
$ =2^{n}.{{\cos }^{n}}\frac{x}{2}.\sin \frac{nx}{2} $
BETA
