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Let $f(n)$ be the function defined by

$$ f(n) = \frac 1{\sqrt{5}} \left[ \left(1+\sqrt{5}\over2\right)^n- \left(1-\sqrt{5}\over2\right)^n \right] $$

How do you prove that $f(n) = f(n+2) - f(n+1)$ for all $n = 0, 1, 2, ...$ ?

Git Gud
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apple
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    You can also prove that $f(n)$ is the $n$-th Fibonacci number, which is essentially what you're saying (I'm assuming you meant $f(n)=f(n+2)-f(n+1)$). This probably simplifies things. – Reveillark Mar 01 '15 at 23:07
  • The $s are your friends – Ben Grossmann Mar 01 '15 at 23:08
  • What do you know about induction? What have you tried? This website is for answering questions, not giving lectures. Do you have to use induction, or can other approaches be used? – DanielV Mar 01 '15 at 23:09
  • Thanks for the editing aid @Omnomnomnom – apple Mar 01 '15 at 23:10
  • I know that you have to start from the base case by proving the given statement for the first natural number. Then, from step case, it needs to be proven that the given statement for any one natural number implies the given statement for the next natural number @DanielV – apple Mar 01 '15 at 23:13
  • @iterence Show us how far along you've gotten with that, otherwise this question will probably be closed. – DanielV Mar 02 '15 at 00:04

1 Answers1

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Hint: note that $x = \frac 12 (1 \pm \sqrt{5})$ satisfies $$ x^2 - x - 1=0 $$ or, more generally, $$ x^n = x^{n+2} - x^{n+1} $$

Ben Grossmann
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