Let's break down the problem systematically. The function $f(n)$ is defined by the following:

• $f(1) = 3$
• $f(n+1) = f(n)^2 + 1$

We need to find the value of $f(4)$.

Step 1: Calculate $f(2)$

Using the recurrence relation: $f(2) = f(1)^2 + 1 = 3^2 + 1 = 9 + 1 = 10$

Step 2: Calculate $f(3)$

$f(3) = f(2)^2 + 1 = 10^2 + 1 = 100 + 1 = 101$

Step 3: Calculate $f(4)$

$f(4) = f(3)^2 + 1 = 101^2 + 1$

Now, calculate $101^2$: $101^2 = 10201$ So: $f(4) = 10201 + 1 = 10202$

Final Answer:

The value of $f(4)$ is $\boxed{10202}$.

Would you like more details or have any questions? Here are some related questions:

1. What pattern do you observe in the values of $f(n)$?
2. How would the values change if $f(1)$ were different?
3. Can you generalize a formula for $f(n)$?
4. What would $f(5)$ be?
5. How does the recurrence relation affect the growth of $f(n)$?

Tip: In recurrence relations, always calculate a few initial terms to understand the pattern and behavior of the function.