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Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
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including olympiad champions, researchers, and professionals.
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?
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Concept Quizzes
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 \( x_5? \)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
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 \( x_n. \)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
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 \( x_n. \)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
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 \( x_6? \)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
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 \( \frac{b}{a}? \)