1. Given the recurrence a_k = 3a_{k – 1} – 2.25a_{k – 2}, a₀ = 0, a₁ = 2.

1. Write out the first four terms of this sequence.
2. What is the characteristic equation?
3. Given the roots to the characteristic equation are both t = 1.5, write an equation for a_n based on the linear combination of the roots to this characteristic equation using real constants C and D.
4. Use the initial conditions to solve for the constants C and D and write an explicit formula for the recurrence relation.
5. Show that your formula works for a₂ by comparison to the list you made in a.

2. Given the recurrence a_k = 7a_{k – 1} – 10a_{k – 2}, a₀ = 2, a₁ = 2.

1. Write out the first four terms of this sequence.
2. What is the characteristic equation?
3. Given the roots to the characteristic equation are t = 2, 5, write an equation for a_n based on the linear combination of the roots to this characteristic equation using real constants C and D.
4. Use the initial conditions to solve for the constants C and D and write an explicit formula for the recurrence relation.
5. Show that your formula works for a₂ by comparison to the list you made in a.

Lemma 8.3.1 Characteristic Equation

Theorem 8.3.3 Distinct Roots Theorem

Theorem 8.3.5 Single-Root Theorem

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