1. ## Fibonacci

{ $a_0,a_1,a_2,....,a_n,....$} := {1,1,2,3,5,8,13,21,34,...}

What is a generic term $a_n$ in this sequence using a 2 X 2 matrix??

What is a formula for $a_n$that only depends on $n$, where n ≥ 3 is a positive integer ???

Any idea for this? I don't know how can I start this??

Thanks.

2. Hello, noppawit!

$a_0,a_1,a_2, \hdots ,a_n, \hdots \;=\;\{1,1,2,3,5,8,13,21,34,\hdots\}$

What is a generic term $a_n$ in this sequence using a 2×2 matrix?
. . Sorry, I don't understand this.

What is a formula for $a_n$ that only depends on $n$, where n ≥ 3 is a positive integer ?

What course are you in?

The closed form for the Fibonacci Sequence is: . $a_n \;=\;\frac{(1+\sqrt{5})^n - (1-\sqrt{5})^n} {2^n\sqrt{5}}$

Its derivation requires rigorous algebra.

3. Hello noppawit
Originally Posted by noppawit
{ $a_0,a_1,a_2,....,a_n,....$} := {1,1,2,3,5,8,13,21,34,...}

What is a generic term $a_n$ in this sequence using a 2 X 2 matrix??

What is a formula for $a_n$that only depends on $n$, where n ≥ 3 is a positive integer ???

Any idea for this? I don't know how can I start this??

Thanks.
There's a whole lot on the Fibonacci sequence, including a matrix form, on Wikipedia here: here

4. Originally Posted by noppawit
{ $a_0,a_1,a_2,....,a_n,....$} := {1,1,2,3,5,8,13,21,34,...}

What is a generic term $a_n$ in this sequence using a 2 X 2 matrix??
...
Check the online encyclopedia of integer sequences.

Fibonacci numbers:
F(0)= 0, F(1)= 1, F(2)= 1, F(3)= 2, F(4)= 3

The matrix representation

$\begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^n =
\begin{bmatrix} F_{n+1} & F_n \\
F_n & F_{n-1} \end{bmatrix}$

or this
$\begin{bmatrix} 0 & 1 \\ 1 & 1 \\ \end{bmatrix}^n \, = \, \begin{bmatrix} F_{n-1} & F_n \\ F_n & F_{n+1} \\ \end{bmatrix}$

$\begin{bmatrix} 0 & 1 \\ 1 & 1 \\ \end{bmatrix}^4 \, = \, \begin{bmatrix} 2 & 3 \\ 3 & 5 \\ \end{bmatrix}$

$\begin{bmatrix} 0 & 1 \\ 1 & 1 \\ \end{bmatrix}^9 \, = \, \begin{bmatrix} 21 & 34 \\ 34 & 55 \\ \end{bmatrix}$

$\begin{bmatrix} 0 & 1 \\ 1 & 1 \\ \end{bmatrix}^{20} \, = \, \begin{bmatrix} 4181 & 6765 \\ 6765 & 10946 \\ \end{bmatrix}$

$\begin{bmatrix} 0 & 1 \\ 1 & 1 \\ \end{bmatrix}^{30} \, = \, \begin{bmatrix} 514229 & 832040 \\ 832040 & 1346269 \\ \end{bmatrix}$

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