If f, of, 1, equals, 2*f*(1)=2 and f, of, n, equals, f, of, n, minus, 1, squared, plus, 4*f*(*n*)=*f*(*n*−1)2+4 then find the value of f, of, 3*f*(3).

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?

---

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.

Learn how to calculate f(3) using recursive functions with a detailed step-by-step solution. This problem involves understanding recursive function definitions and applying them to find specific values in sequences. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation