Question

If $x_r=\cos \left(\frac{\pi }{3^r}\right)+i\sin \left(\frac{\pi }{3^r}\right)$ then $x_1\cdot x_2\cdot x_3...\infty$ equals

• A. $-i$
• B. $i$
• C. $-1$
• D. $1$

Answer 1

Answer: (B)

$i$

$x_r=\cos \left(\frac{\pi }{3^r}\right)+i\sin \left(\frac{\pi }{3^r}\right)=e^{\frac{i\pi }{3^r}}={\left(e^{i\pi }\right)}^{\frac{1}{3^r}}$
Let $y=e^{i\pi }$
$\Rightarrow x_r=y^{\frac{1}{3^r}}$
$\Rightarrow x_1{x}_2{x}_3......\infty =y^{(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{27}+...+\infty )}=y^{\left(\frac{\frac{1}{3}}{1-\frac{1}{3}}\right)}=y^{\frac{1}{2}}$
$\Rightarrow x_1{x}_2{x}_3......\infty =e^{\frac{i\pi }{2}}=i$
Hence, option 'B' is correct.