Question

$\sqrt{2}+\sqrt{2}+\sqrt{2}+...+nterms=2\cos \frac{\pi }{2^{n+1}},nϵN$

Answer 1

Using Mathematical inducion

$Step\left(I\right)=n=1$

$P\left(1\right)=\sqrt{2}=2cos\frac{\pi }{2^2}=2\cdot \frac{1}{\sqrt{2}}=\sqrt{2}$

$\therefore P\left(1\right)$ holds .

$step\left(II\right)$ = Assume that $P\left(n\right)$ is true

$P\left(n\right)=2cos\frac{\pi }{2^n+1}$

$Step\left(III\right)=$ $P(n+1)=\sqrt{2+P\left(n\right)}=\sqrt{2+2cos\frac{\pi }{2^{n+1}}}$

$=\sqrt{2(1+cos\frac{\pi }{2^{n+1}})}=\sqrt{2\cdot 2co{s}^2\frac{\pi }{2^{n+2}}}$

$=2cos\frac{\pi }{2^{n+2}}$

Thus $P(n+1)$ holds good.

$\therefore P$ holds for every natural number $n$.