Probability

Linear Recurrence Relations

Linear Recurrence Relations - Basic Substitutions

         

The sequence {xn} \{x_n\} satisfies the relation x1=5,xn+1=xn+5(n=1,2,3,…). \begin{array}{c}&x_1 = 5,&x_{n+1} = x_n + 5 &(n = 1, 2, 3, \ldots). \end{array} What is the value of x20? x_{20}?

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The sequence {xn} \{x_n\} satisfies the relation x1=2,xn+1=2xn+1(n=1,2,3,…). \begin{array}{c}&x_1 = 2, &x_{n+1} = 2 x_n + 1 &(n = 1, 2, 3, \ldots). \end{array} What is the value of x5? x_{5}?

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The sequence {xn} \{x_n\} satisfies the relation x1=1,x2=5,xn+2=4xn+1+5xn(n=1,2,3,…). \begin{array}{c}&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 x8625? \dfrac{x_8}{625}?

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The sequence {xn} \{x_n\} satisfies the relation x1=1,x2=0,6xn+2=5xn+1−xn(n=1,2,3,⋯ ). \begin{array}{c}&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 n that satisfies xn=1310−1210? x_n = \dfrac{1}{3^{10}} - \dfrac{1}{2^{10}}?

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The sequence {xn} \{x_n\} satisfies the relation x1=4,xn+1=xn+3n−2(n=1,2,3,…).\begin{array}{c}&x_1 = 4, &x_{n+1} = x_n + 3n - 2 &(n = 1, 2, 3, \ldots). \end{array} What is the value of x12? x_{12}?

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