Final answer:
To prove an explicit formula for a recurrence relation, we use mathematical induction, checking the base case and assuming the formula holds for c_k and c_k-1 to then prove it for c_k+1.
Step-by-step explanation:
The question involves proving that a given explicit formula solves a recurrence relation. The recurrence relation given is cℓ = 4cℓ₋₁ - 4cℓ₋₂ with initial conditions c0 = -3 and c1 = 4. We need to show that the explicit formula cn = -3(2n) + 5n(2n) holds true for this recurrence relation for all n ≥ 0.
To prove this, we can use mathematical induction. For a base case, we substitute n=0 and n=1 into our explicit formula to check if they satisfy the initial conditions. After the base case is confirmed, we assume the formula holds for ck and ck-1, and then we prove it also holds for ck+1 following the recurrence relation. This involves substituting ck and ck-1 with their respective explicit expressions in the relation and simplifying to show that the resultant expression aligns with the explicit formula for ck+1.