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 May 12th, 2012, 11:26 PM #1 Newbie   Joined: Mar 2012 Posts: 6 Thanks: 0 Recurrence relation For the recurrence relation $x_{xn-1}= 5_{xn-1} - 6_{xn-2}; for\ n\geq2 x_{1}=1 x_{0}=0$ prove by induction that: $\begin{bmatrix} x_{n}\\ x_{n-1} \end{bmatrix} = \begin{bmatrix} 5 &-6 \\ 1 & 0 \end{bmatrix}^{n-1}\begin{bmatrix} 1\\0 \end{bmatrix}$
 May 13th, 2012, 06:17 AM #2 Senior Member   Joined: Jul 2011 Posts: 227 Thanks: 0 Re: Recurrence relation What have you tried?
May 15th, 2012, 10:49 AM   #3
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Re: Recurrence relation

Hello, Dragonkiller!

Quote:
 $\text{For the recurrence relation : }\:x_{n}\:=\: 5x_{n-1}\,-\,6x_{n-2}\;\text{ for }n\,\geq\,2,\;\;x_0 \,=\,0,\;x_1\,=\,1$ [color=beige]. . [/color]$\text{prove by induction that: }\:\begin{bmatrix}x_{n} \\ x_{n-1} \end{bmatrix} \:=\: \begin{bmatrix} 5 &-6 \\ 1=&0 \end{bmatrix}^{n-1}\begin{bmatrix} 1\\0 \end{bmatrix}=$

I haven't come up with the inductive proof yet.

But I have the general term:[color=beige] .[/color]$\begin{Bmatrix}x_0=&0 \\ x_1=&1 \\ x_{n}=&3^n - 2^n \end{Bmatrix}=$

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