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Recursion is a method in programming and mathematics where a function calls itself to solve smaller instances of the same problem. This approach can be very powerful for breaking down complex problems into manageable pieces.
In the given exercise, recursion is used to define the function \(f(n)\) in terms of itself for smaller values of \(n\). The recursive formula is \(f(n) = f(n / 5) + 3n^2\), and it helps in calculating \(f(n)\) for larger integers by successively breaking them down.

Recursion generally involves two main parts:

• The Recursive Case: This is the part of the function that calls itself with a smaller or simpler input.
• The Base Case: This is the condition upon which the function terminates without calling itself. It prevents infinite loops.

Recursion can make solving problems intuitive, as you only need to specify how to break the problem into smaller parts and when to stop.

The base case is a fundamental concept in recursion that helps prevent infinite loops.
It provides a simple, non-recursive solution to a small part of the problem. In our exercise, the base case is given as \(f(1) = 4\). This means that when \(n = 1\), the value of \(f(n)\) is 4, and no further recursion is necessary.

The base case acts like an anchor that stabilizes the recursive process. Without a base case, the function would keep calling itself endlessly. Here is why the base case is crucial:

• Termination: It ensures that the recursive calls end at some point.
• Solution Buildup: Provides a starting point to build and accumulate intermediate solutions.

Knowing the base case allows us to calculate \(f(n)\) for larger values by drawing back to these simple known values. For example, in our problem, the base case is used to find \(f(5)\), which is in turn used to find \(f(25)\), and so on.

A step-by-step solution helps in understanding how to arrive at the final answer through a series of intermediate steps. Let's review how it works in our exercise.

Step 1: Understand the Base Case
The base case is \(f(1) = 4\). This is our starting point for calculating larger values.
Step 2: Find \(f(5)\)
Use the recursive formula: \(f(5) = f(5/5) + 3 \times 5^2\).
We already know that \(f(1) = 4\), so:
\(f(5) = 4 + 3 \times 25 = 4 + 75 = 79\).

Step 3: Find \(f(125)\)
First, calculate \(f(25)\):
\(f(25) = f(25/5) + 3 \times 25^2 = 79 + 3 \times 625 = 79 + 1875 = 1954\).
Then use this to find:
\(f(125) = 1954 + 3 \times 15625 = 1954 + 46875 = 48829\).

Step 4: Find \(f(3125)\)
Calculate \(f(625)\) first:
\(f(625) = f(625/5) + 3 \times 625^2 = 48829 + 3 \times 390625 = 48829 + 1171875 = 1220704\).
Finally, use it to find:
\(f(3125) = 1220704 + 3 \times 9765625 = 1220704 + 29296875 = 30517579\).
The step-by-step approach helps break down the problem and ensure that we apply the recursive formula correctly at each stage.