Find f(1), f(2), f(3), and, f(4) if f(n) is defined recursively by f(0) = 1 and for n = 0, 1, 2, . . .

1

ICS 141: Discrete Mathematics I (Fall 2014)

a) f(n + 1) = f(n) + 2

f(1) = f(0 + 1) = f(0) + 2 = 1 + 2 = 3

f(2) = f(1 + 1) = f(1) + 2 = 3 + 2 = 5

f(3) = f(2 + 1) = f(2) + 2 = 5 + 2 = 7

f(4) = f(3 + 1) = f(3) + 2 = 7 + 2 = 9

b) f(n + 1) = 3f(n)

f(1) = f(0 + 1) = 3f(0) = 3(1) = 3

f(2) = f(1 + 1) = 3f(1) = 3(3) = 9

f(3) = f(2 + 1) = 3f(2) = 3(9) = 27

f(4) = f(3 + 1) = 3f(3) = 3(27) = 81

Let's break down the two parts of the problem:

### Part (a): Recursive function \( f(n+1) = f(n) + 2 \)

We are given the initial condition \( f(0) = 1 \) and the recursive relation \( f(n+1) = f(n) + 2 \). Now, let's find \( f(1), f(2), f(3), \) and \( f(4) \).

1. \( f(1) = f(0) + 2 = 1 + 2 = 3 \)
2. \( f(2) = f(1) + 2 = 3 + 2 = 5 \)
3. \( f(3) = f(2) + 2 = 5 + 2 = 7 \)
4. \( f(4) = f(3) + 2 = 7 + 2 = 9 \)

So, the values are:
- \( f(1) = 3 \)
- \( f(2) = 5 \)
- \( f(3) = 7 \)
- \( f(4) = 9 \)

### Part (b): Recursive function \( f(n+1) = 3f(n) \)

We are given the initial condition \( f(0) = 1 \) and the recursive relation \( f(n+1) = 3f(n) \). Now, let's find \( f(1), f(2), f(3), \) and \( f(4) \).

1. \( f(1) = 3f(0) = 3 \times 1 = 3 \)
2. \( f(2) = 3f(1) = 3 \times 3 = 9 \)
3. \( f(3) = 3f(2) = 3 \times 9 = 27 \)
4. \( f(4) = 3f(3) = 3 \times 27 = 81 \)

So, the values are:
- \( f(1) = 3 \)
- \( f(2) = 9 \)
- \( f(3) = 27 \)
- \( f(4) = 81 \)

---

Let me know if you'd like more details or have questions.

Here are 5 related questions to deepen understanding:
1. What would \( f(5) \) be for both part (a) and part (b)?
2. How would the function change in part (a) if the recursive relation was \( f(n+1) = f(n) + 3 \)?
3. What is the general formula for \( f(n) \) for part (a)?
4. Can you find the general formula for \( f(n) \) in part (b)?
5. How does changing the initial condition \( f(0) \) affect the results in both parts?

**Tip:** Recursion is often simplified by identifying patterns, which can sometimes lead to discovering a closed-form solution!

Find f(1), f(2), f(3), and f(4) if f(n) is defined recursively by f(0) = 1 and for n = 0, 1, 2, . . .
a) f(n + 1) = f(n) + 2
b) f(n + 1) = 3f(n)

Solve a recursive sequence problem where f(n) is defined by two recursion rules: f(n+1) = f(n) + 2 and f(n+1) = 3f(n), with the initial condition f(0) = 1. Learn how to calculate f(1), f(2), f(3), and f(4) step-by-step.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation