https://file.aiquickdraw.com/mathbot/file/1738636792308t1i040oq.jpg

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! 😊

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### 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!

If f(1) = 3 and f(n) = f(n-1)^2 + 5, then find the value of f(4).

Learn how to find the value of f(4) for a recursive function defined as f(1) = 3 and f(n) = f(n-1)^2 + 5. Detailed step-by-step calculations show the recursive process. Suitable for high school students studying recursion in algebra.

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