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For each $n$ consider the expression $$\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^n\right]$$

I am trying to prove by induction that this is an integer for all $n$.

In the base case $n=1$, it ends up being $1$.

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I am trying to prove the induction step:

- if $\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^n\right]$ is an integer, then so is

$\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right]$.

I have tried expanding it, but didn’t get anywhere.

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Try writing

$$

\left(\frac{1+\sqrt{5}}{2}\right)^n = \frac{a_n+b_n\sqrt{5}}{2}

$$

with $a_1=b_1=1$ and

$$

\frac{a_{n+1}+b_{n+1}\sqrt{5}}{2} = \left(\frac{1+\sqrt{5}}{2}\right)\left(\frac{a_n+b_n\sqrt{5}}{2}\right)

$$

see what you get, then repeat as much as you need to with $\left(\frac{1-\sqrt{5}}{2}\right)^n$.

**Hint** $\rm\quad \phi^{\:n+1}\!-\:\bar\phi^{\:n+1} =\ (\phi+\bar\phi)\ (\phi^n-\:\bar\phi^n)\ -\ \phi\:\bar\phi\,\ (\phi^{\:n-1}\!-\:\bar\phi^{\:n-1})$

Therefore, upon substituting $\rm\ \phi+\bar\phi\ =\ 1\ =\, -\phi\bar\phi\ $ and dividing by $\:\phi-\bar\phi = \sqrt 5\:$ we deduce that $\rm\:f_{n+1} = f_n + f_{n-1}.\:$ Since $\rm\:f_0,f_1\:$ are integers, all $\rm\,f_n\:$ are integers by induction, using the recurrence.

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