May someone help me? I am trying to use induction to prove that the formula for finding the $n^{th}$ term of the Fibonacci sequence is:
$$F_n=\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^n-\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^n$$
I tried to put $n=1$ into the equation and prove that if $n=1$ works then $n=2$ works and it should work for any number but it didn't work. I need to prove that this formula gives the $n^{th}$ Fibonacci number.