We are given the following recursive function:

1. $f(1) = 3$
2. $f(n) = f(n-1)^2 + 5$, for $n > 1$.

We need to find $f(4)$. Let us calculate step by step:

Step 1: Calculate $f(2)$

Using the recurrence relation: $f(2) = f(1)^2 + 5$ Substituting $f(1) = 3$: $f(2) = 3^2 + 5 = 9 + 5 = 14$

Step 2: Calculate $f(3)$

Using the recurrence relation: $f(3) = f(2)^2 + 5$ Substituting $f(2) = 14$: $f(3) = 14^2 + 5 = 196 + 5 = 201$

Step 3: Calculate $f(4)$

Using the recurrence relation: $f(4) = f(3)^2 + 5$ Substituting $f(3) = 201$: $f(4) = 201^2 + 5 = 40401 + 5 = 40406$

Final Answer:

$f(4) = 40406$

Let me know if you'd like a detailed explanation or have additional questions! 😊

Related Questions:

1. What are some real-world applications of recursive functions like this one?
2. Can this recurrence relation be solved explicitly for general $n$?
3. How does the growth rate of $f(n)$ compare to exponential or factorial functions?
4. What happens if we redefine $f(n)$ with subtraction instead of addition in the recurrence?
5. How can this recurrence relation be programmed in Python?

Tip:

When solving recursive problems, always start with the base case and systematically compute the terms step by step!