Final answer:
To prove that the explicit formula aₙ = 1/2 + 1/2*3ⁿ is the solution to the given recurrence relation, we will use mathematical induction. By substituting the explicit formula into the recurrence relation, we can show that it holds true for any value of n. Therefore, we can conclude that the explicit formula is a solution to the recurrence relation.
Step-by-step explanation:
To prove that the explicit formula aₙ = 1/2 + 1/2*3ⁿ is the solution to the given recurrence relation, we will use mathematical induction.
Step 1: Base Case
Substituting n = 0 into the explicit formula, we have:
a₀ = 1/2 + 1/2*3⁰ = 1/2 + 1/2*1 = 1/2 + 1/2 = 1.
Substituting n = 1 into the explicit formula, we have:
a₁ = 1/2 + 1/2*3¹ = 1/2 + 1/2*3 = 1/2 + 3/2 = 2.
Since the base cases match the given initial conditions, the base case is satisfied.
Step 2: Inductive Step
Assuming the explicit formula holds for some value k, we will now prove that it holds for k + 1.
By the recurrence relation, aₙ = 4aₙ₋₁ - 3aₙ₋₂.
Plugging in aₖ for aₙ₋₁ and aₖ₋₁ for aₙ₋₂, we get:
aₖ₊₁ = 4aₖ - 3aₖ₋₁
Substituting the explicit formula for aₖ and aₖ₋₁, we have:
aₖ₊₁ = 4(1/2 + 1/2*3ᵏ) - 3(1/2 + 1/2*3ˣ⁻¹)
Simplifying the expression, we get:
aₖ₊₁ = 2 + 2*3ᵏ - 3/2 - 3/2*3⁻¹
aₖ₊₁ = 2 - 3/2*3⁻¹ + 2*3ᵏ - 3/2*3ᵏ
Factoring out 3ᵏ terms, we have:
aₖ₊₁ = 1/2 + 1/2*3^k * (2*3 - 3/2)
aₖ₊₁ = 1/2 + 1/2*3⁰ * 3
aₖ₊₁ = 1/2 + 1/2*3ˡ
This expression is the same as the explicit formula when n = k + 1.
Step 3: Conclusion
Since the base case is satisfied and the inductive step holds true, we can conclude that the explicit formula aₙ = 1/2 + 1/2*3ⁿ is a solution to the given recurrence relation.