1. Given the recurrence a_k = -a_k - 1 + 12a_k - 2, a₀ = 1, a₁ = 8.

1. Write out the next three terms of this sequence.
2. What is the characteristic equation?
3. Given the roots to the characteristic equation are r = 3 and s = -4, 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.
   

2. Given the recurrence a_k = 10a_k - 1 - 25a_k - 2, a₀ = 0, a₁ = 1.

1. Write out the next three terms of this sequence.
2. What is the characteristic equation?
3. Given the roots to the characteristic equation both t = 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.