Question

Find all solutions to the recurrence relation aₙ=−5ₐₙ₋₁−6ₐₙ−2+42⋅4ⁿ with the initial conditions ₐ₁=56 and ₐ₂=278. (without using the generating functions)

Answer 1

To find all solutions to the given recurrence relation, use the back substitution method to find subsequent terms of the sequence.

Step-by-step explanation:

To find all solutions to the given recurrence relation, we can use a method called back substitution. Let's first write out the first few terms of the sequence:

aₙ = -5aₙ₋₁ - 6aₙ₋₂ + 42 × 4ⁿ

a₁ = 56

a₂ = 278

We can start by finding a₃ using the given relation:

a₃ = -5(278) - 6(56) + 42 × 4³

= -1390 - 336 + 672

= -1054

Continuing in this manner, we can find subsequent terms of the sequence until we reach the desired number of terms.