Discrete Mathematics

Linear Recurrence Relations

Linear Recurrence Relations - With Repeated Roots

         

The sequence {xn} \{x_n\} satisfies the relation x1=2,x2=9,xn+2=6xn+19xn(n=1,2,3,). \begin{array}{c}&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 x5? x_5?

The sequence {xn} \{x_n\} satisfies the relation x1=3,x2=9,x3=29,xn+3=9xn+226xn+1+24xn, \begin{array}{c}&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 nn is a positive integer. Find the closed-form expression of xn. x_n.

The sequence {xn} \{x_n\} satisfies the relation x1=3,x2=9,x3=9,xn+3=9xn+227xn+1+27xn,\begin{array}{c}&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 nn is a positive integer. Find the closed-form expression of xn. x_n.

The sequence {xn} \{x_n\} satisfies the relation x1=7,x2=0,x3=6,xn+3=2xn+2+5xn+16xn, \begin{array}{c}&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 nn is a positive integer. What is the value of x6? x_6?

The sequence {xn} \{x_n\} satisfies the relation x1=0,x2=1,xn+2+2xn+1+xn=0(n=1,2,3,). \begin{array}{c}&x_1 = 0, &x_2 = 1, &x_{n+2} + 2 x_{n+1} + x_n = 0 &(n = 1, 2, 3, \ldots). \end{array} If xn x_n can be expressed as arn+bnrn, a r^n + b n r^n, what is the value of ba? \frac{b}{a}?

×

Problem Loading...

Note Loading...

Set Loading...