If f, of, 1, equals, 1*f*(1)=1 and f, of, n, equals, f, of, n, minus, 1, squared, plus, n*f*(*n*)=*f*(*n*−1)2+*n* then find the value of f, of, 3*f*(3).

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.

Learn how to calculate the recursive function f(n) = f(n-1)^2 + n with step-by-step solutions. Explore the value of f(3) and understand the recursive relationship. Suitable for students in Grades 10-12.

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