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Linear Recurrence Relations

Take your recursion skills to the next level. If you've got a recurrence relation but no computer, how can you find a closed form? What about asymptotic behavior? How fast do rabbits reproduce?

Repeated Roots

         

The sequence \( \{x_n\} \) satisfies the relation \[ \begin{array} &x_1 = 2, &x_2 = 9, &x_{n+2} = 6 x_{n+1} - 9 x_n &(n = 1, 2, 3, \ldots). \end{array}\] What is the value of \( x_5? \)

The sequence \( \{x_n\} \) satisfies the relation \[ \begin{array} &x_1 = 3, &x_2 = 9, &x_3 = 29, &x_{n+3} = 9 x_{n+2} - 26 x_{n+1} + 24 x_n, \end{array}\] where \(n\) is a positive integer. Find the closed-form expression of \( x_n. \)

The sequence \( \{x_n\} \) satisfies the relation \[\begin{array} &x_1 = -3, &x_2 = -9, &x_3 = 9, &x_{n+3} = 9 x_{n+2} - 27 x_{n+1} + 27 x_n, \end{array} \] where \(n\) is a positive integer. Find the closed-form expression of \( x_n. \)

The sequence \( \{x_n\} \) satisfies the relation \[ \begin{array} &x_1 = -7, &x_2 = 0, &x_3 = 6, &x_{n+3} = 2 x_{n+2} + 5 x_{n+1} - 6 x_n, \end{array}\] where \(n\) is a positive integer. What is the value of \( x_6? \)

The sequence \( \{x_n\} \) satisfies the relation \[ \begin{array} &x_1 = 0, &x_2 = 1, &x_{n+2} + 2 x_{n+1} + x_n = 0 &(n = 1, 2, 3, \ldots). \end{array}\] If \( x_n \) can be expressed as \[ a r^n + b n r^n, \] what is the value of \( \frac{b}{a}? \)

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