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Recently I am working on recurrence relation and I have gathered the following.

  1. Fibonacci sequence
  2. Solving linear homogeneous recurrence relation by characteristics equation

So I attempted to work on the closed form of Fibonacci sequence by myself. However, it seems to contradict to another source attached below indicating the closed form of Fibonacci sequence. May I know what's wrong?

Solution technique:

  1. Establish the characteristics equation and its roots. $$r^2-r-1=0$$ $$r=\frac{1\pm\sqrt5}{2}$$
  2. Assume $f_n=c_1r_1^n+c_2r_2^n$, where $r_1$ and $r_2$ are distinct roots in this case.
  3. Find $c_1,\;c_2$ $$r_1=\frac{1+\sqrt5}{2},\;r_2=\frac{1-\sqrt5}{2}$$ $$f_0=c_1r_1^0+c_2r_2^0=c_1+c_2=1$$ $$f_1=c_1r_1^1+c_2r_2^1=c_1r_1+c_2r_2=2$$ $$\left[\begin{array}{cc|c}1&1&1\\\frac{1+\sqrt5}{2}&\frac{1-\sqrt5}{2}&2\end{array}\right]=\cdots=\left[\begin{array}{cc|c}1&0&\frac{5+3\sqrt5}{10}\\0&1&\frac{5-3\sqrt5}{10}\end{array}\right] $$ $c_1\neq\frac{1}{\sqrt5},\;c_2\neq-\frac{1}{\sqrt5}$ as indicated in the below closed form, what's wrong in my derivation?

The closed form I would like to replicate using characteristic equation

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  • $\begingroup$ What number is $f_1$? $\endgroup$ Feb 17 '20 at 4:23
  • $\begingroup$ I am not sure whether is $f_0=1$ or $f_0=0$...May I know the convention also? $\endgroup$
    – Andes Lam
    Feb 17 '20 at 4:24
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    $\begingroup$ Usually $f_0$ is taken to be $0$ and $f_1=f_2=1$ $\endgroup$ Feb 17 '20 at 4:25
  • $\begingroup$ I think we should have $\{\frac{1}{\sqrt{5}}⋅\left(\frac{1+\sqrt{5}}{2}\right)^n,\frac{1}{\sqrt{5}}⋅\left(\frac{1-\sqrt{5}}{2}\right)^n\}$ with $n=2$ in the first row and $n=3$ in the second row $\endgroup$
    – Matt Groff
    Feb 17 '20 at 4:29
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The usual convention is $f_0=0, f_1=f_2=1$. Using your equations $$c_1+c_2=0\\c_1r_1+c_2r_2=1\\ c_1\frac{1+\sqrt 5}2+c_2\frac{1-\sqrt 5}2=1\\ \frac {\sqrt 5}2(c_1-c_2)=1\\ c_1=\frac 1{\sqrt 5}\\c_2=-\frac 1{\sqrt 5}$$

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