Practice - Characterization of Sequences
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Practice Questions
Test your understanding with targeted questions
What is a linear homogeneous recurrence equation?
💡 Hint: Look for terms that establish a relationship between sequence terms.
Define characteristic roots.
💡 Hint: Think of them as the values essential for the general solution.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What is the characteristic equation for the recurrence $a_n = a_{n-1} + a_{n-2}$?
💡 Hint: Think of how each previous term influences the next.
True or False: A linear homogeneous recurrence can have non-constant terms.
💡 Hint: Recall what it means for an equation to be homogeneous.
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Challenge Problems
Push your limits with advanced challenges
Given the recurrence $a_n = 2a_{n-1} + a_{n-2}$ with initial conditions $a_0 = 1$, $a_1 = 2$, derive and solve for $a_2$ and $a_3$. Show each step.
💡 Hint: Remember to substitute the previous terms into the equation.
Prove that if an equation has distinct roots, the form $a_n = \alpha r_1^n + \beta r_2^n$ holds true. Use an example.
💡 Hint: You might require the quadratic formula to find your roots.
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