Given the recursive function definition:

$f(1) = 2$

$f(n) = f(n - 1)^2 + 4$

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

Step 1: Calculate $f(2)$

Using the recursive formula: $f(2) = f(1)^2 + 4$ Substitute $f(1) = 2$: $f(2) = 2^2 + 4 = 4 + 4 = 8$

Step 2: Calculate $f(3)$

Using the recursive formula: $f(3) = f(2)^2 + 4$ Substitute $f(2) = 8$: $f(3) = 8^2 + 4 = 64 + 4 = 68$

So, the value of $f(3)$ is $\boxed{68}$.

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

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Here are 5 related questions:

1. What would be the value of $f(4)$ using the same recursive formula?
2. How does changing the constant term from 4 to another value affect the sequence?
3. Can you derive a general formula for $f(n)$ using the given recursion?
4. How would the sequence change if the initial condition were $f(1) = 1$ instead of $f(1) = 2$?
5. What is the relationship between the recursion formula and quadratic growth?

Tip: In recursive sequences, small changes in initial conditions or the recursion rule can lead to significant differences in the sequence, which is essential to explore.