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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?

# Linear Recurrence Relations - 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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