Recursive sequences are sequences of numbers which can be represented by recursive formula. The most famous example continues is a Fibonacci Sequence. Consider the following sequence {fₙ | n ≥ 0} defined as: f₀ = 1, f₁ = 1, f₂ = 2 and, for i > 2, define fᵢ := 2fᵢ₋₁ + 7fᵢ₋₂ + 4fᵢ₋₃.
(a) A common strategy with such recursive sequences is view it as a matrix multipli- fi cation.
Let ( fᵢ )
( fᵢ₋₁ )
( fᵢ₋₂ )
Then, the recurrence relation can be represented as:
Specifically, the relation can be represented as an equation of the following form:
A · Fᵢ₋₁ = Fᵢ (Or) A . ( fᵢ₋₁ ) = ( fᵢ )
( fᵢ₋₂ ) ( fᵢ₋₁ )
( fᵢ₋₃ ) ( fᵢ₋₂ )
What is the satisfying value of A? Also, derive an equation for Fᵢ in terms of F₂, A and i. Justify your answers.
