Given the recursive function definition:

$f(1) = 1$

$f(n) = f(n - 1)^2 + n$

We need to find the value of $f(3)$.

Step 1: Calculate $f(2)$

Using the recursive formula: $f(2) = f(1)^2 + 2$ Substitute $f(1) = 1$: $f(2) = 1^2 + 2 = 1 + 2 = 3$

Step 2: Calculate $f(3)$

Again using the recursive formula: $f(3) = f(2)^2 + 3$ Substitute $f(2) = 3$: $f(3) = 3^2 + 3 = 9 + 3 = 12$

So, the value of $f(3)$ is $\boxed{12}$.

Would you like to see more details or have any further questions?

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Here are 5 related questions:

1. How would you generalize this recursive function for any value of $n$?
2. What is the value of $f(4)$ using the same recursive formula?
3. Can you express the recursive formula in terms of an iterative process?
4. What would happen if you change the recursive relation to $f(n) = 2f(n-1) + n$?
5. How does the initial condition $f(1) = 1$ affect the overall sequence?

Tip: When solving recursive functions, it’s helpful to work out the first few terms manually to observe any patterns that might emerge.