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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?

Basic Substitutions

The sequence \( \{x_n\} \) satisfies the relation \[ \begin{array} &x_1 = 5,&x_{n+1} = x_n + 5 &(n = 1, 2, 3, \ldots). \end{array} \] What is the value of \( x_{20}? \)

The sequence \( \{x_n\} \) satisfies the relation \[ \begin{array} &x_1 = 2, &x_{n+1} = 2 x_n + 1 &(n = 1, 2, 3, \ldots). \end{array} \] What is the value of \( x_{5}? \)

The sequence \( \{x_n\} \) satisfies the relation \[ \begin{array} &x_1 = 1, &x_2 = 5, &x_{n+2} = 4 x_{n+1} + 5 x_n &(n = 1, 2, 3, \ldots). \end{array}\] What is the value of \( \dfrac{x_8}{625}? \)

The sequence \( \{x_n\} \) satisfies the relation \[ \begin{array} &x_1 = 1, &x_2 = 0, &6x_{n+2} = 5 x_{n+1} - x_n &(n = 1, 2, 3, \cdots). \end{array}\] What is the value \( n \) that satisfies \( x_n = \dfrac{1}{3^{10}} - \dfrac{1}{2^{10}}? \)

The sequence \( \{x_n\} \) satisfies the relation \[\begin{array} &x_1 = 4, &x_{n+1} = x_n + 3n - 2 &(n = 1, 2, 3, \ldots). \end{array} \] What is the value of \( x_{12}? \)

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