Question

If n is a positive integer, prove that (√3+1)^n +(√3-1)^n = 2^n+1 cosπn/6

Answer 1

Given that n is a positive integer.

We have to prove

$( \sqrt{3}+1 )^n +( \sqrt{3}-1)^n= 2^{n+1} cos\frac{\pi n }{6}$

Question is wrong

Take $n=2$ .

$L.H.S=(\sqrt{3}+1)^2+(\sqrt {3} -1)^2$

$=3+1+2\sqrt{3}+3+1-2\sqrt {3}$

$=8$

$R.H.S=2^{2+1} cos\frac{ 2\pi }{6}$ $=2^3×\frac{1}{2}=4$

Hence , $L.H.S \neq R.H.S$