Final answer:
The question deals with solving a second-order homogeneous linear recurrence relation with constant coefficients, using initial conditions to determine specific constants for the general solution.
Step-by-step explanation:
The recurrence relation given is aₙ₊₂ - 7aₙ₊₁ + aₙ = 0, with initial conditions a₀ = 5 and a₁ = 16. To solve this relation, we first need to find its characteristic equation, which is r² - 7r + 1 = 0. Solving this quadratic equation, we get the roots r₁ and r₂, which are the characteristic roots of this recurrence relation.
Once we have the roots, we can express the general solution to the recurrence relation as aₙ = Ar₁ⁿ + Br₂ⁿ, where A and B are constants that can be solved using the initial conditions. Substituting n = 0 and n = 1 into the general solution, we can form a system of equations using the initial conditions provided. Solving for A and B in these equations will give us the specific solution to the recurrence relation.
If r₁ and r₂ are distinct real numbers, then A and B will be determined straightforwardly. If the roots are complex or repeated, additional steps might be needed to express A and B correctly. After finding A and B, we simply substitute them back into the general solution to obtain the terms of the sequence defined by the recurrence relation.