Final answer:
The student's question requires solving a recurrence relation and verifying it with induction. The exact equation is not provided, but typically this involves finding patterns, characteristic equations, and using induction to prove the closed form solution for all natural numbers.
Step-by-step explanation:
The question involves finding a closed form solution for a given recurrence relation and then proving it using induction. The recurrence relation seems to be missing from the prompt; however, typical recurrence relations have the form ak = f(ak-1, ak-2, ...), with specified initial conditions such as a0 = 4. To find a closed form solution, one might look for patterns or use methods such as generating functions or characteristic equations. Once a conjectured closed form is found, mathematical induction can be used to prove it is correct for all natural numbers. The process of induction involves proving the statement is true for the initial case (usually k=0) and then showing if the statement holds for an arbitrary case k, it also holds for the next case k+1.