Final answer:
To solve the recurrence relation a₍ = 5a₍₋₁ + 5ⁿ − 2ⁿ with a₀ = 1, we need to find a pattern or a closed-form expression for a₍, consider the initial condition for constants, and verify the solution satisfies the original relation.
Step-by-step explanation:
The recurrence relation to solve is a₍ = 5a₍₋₁ + 5ⁿ − 2ⁿ, with the initial condition a₀ = 1. To solve the recurrence, we need to find a pattern or a closed-form solution that allows us to calculate a₍ without having to rely on the previous terms.
We can start by trying to see if the non-homogeneous part of the relation, 5ⁿ − 2ⁿ, can be simplified or if there's a pattern we can exploit. If we look at a few terms of the sequence, this might give us a clearer idea of how to approach the problem.
Once we identify any patterns or can create a simpler form, we'll need to consider the initial condition to solve for any constants that may appear in our solution. The final part will involve verifying our solution by plugging it back into the original recurrence relation.