Practice Characterization Of Sequences (14.5) - Solving Linear Homogenous Recurrence Equations – Part I

Practice: Characterization of Sequences

Lesson 14.5Solving Linear Homogenous Recurrence Equations – Part I

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8 questions

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6 sample questions

Question 1.
What is a linear homogeneous recurrence equation?
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Look for terms that establish a relationship between sequence terms.
Question 2.
Define characteristic roots.
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Think of them as the values essential for the general solution.
3Question 3.
What is the characteristic equation for the recurrence $a_n = a_{n-1} + a_{n-2}$ ?
- $r^2 - r - 1 = 0$
- $r^2 - 2 = 0$
- $r^2 + 1 = 0$
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Think of how each previous term influences the next.
4Question 4.
True or False: A linear homogeneous recurrence can have non-constant terms.
- True
- False
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Recall what it means for an equation to be homogeneous.
Question 5.
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.
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Remember to substitute the previous terms into the equation.
Question 6.
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.
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You might require the quadratic formula to find your roots.

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