First slide

Functions (XII)

Question

$If f (x) = \sin x + \tan \frac{x}{2} + \sin \frac{x}{2^{2}} + \tan \frac{x}{2^{3}} + \dots + \sin \frac{x}{2^{n - 1}} + \tan \frac{x}{2^{n}} is a periodic function with period k π, then k =$

Moderate

A

1
B

2
C

$2^{n}$
D

$\frac{1}{2^{n}}$

Solution

$\sin x, \sin \frac{x}{2^{2}}, \dots, \sin \frac{x}{2^{n - 1}} are periodic functions with$
$period 2 π, 2^{3} π, 2^{5} π, \dots, 2^{n} π, respectively and \tan \frac{x}{2}, \tan \frac{x}{2^{3}},$
$\tan \frac{x}{2^{5}}, \dots, \tan \frac{x}{2^{n}} are periodic functions with period 2 π,$
$2^{3} π, 2^{5} π, \dots, 2^{n} π, respectively. L.C.M. of 2 π, 2^{3} π, 2^{5} π, \dots 2^{n} π is 2^{n} π .$
$Hence f (x) is a periodic function with period 2^{n} π .$

$∴ k = 2^{n}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App