You are given the function \( f \), defined recursively as:

- \( f(1) = 2 \)
- \( f(n) = f(n - 1) + 1 \), for \( n > 1 \)

This recursive definition implies that each subsequent value of \( f(n) \) is the previous value incremented by 1. We can deduce the values of \( f(n) \) for some small \( n \):

- \( f(1) = 2 \)
- \( f(2) = f(1) + 1 = 2 + 1 = 3 \)
- \( f(3) = f(2) + 1 = 3 + 1 = 4 \)
- \( f(4) = f(3) + 1 = 4 + 1 = 5 \)
- \( f(5) = f(4) + 1 = 5 + 1 = 6 \)
- And so on...

### General form:
This suggests that the function is increasing linearly. In fact, the recursive rule defines an arithmetic sequence. We can generalize it as:

\[
f(n) = f(1) + (n - 1)
\]

Since \( f(1) = 2 \), we have:

\[
f(n) = 2 + (n - 1) = n + 1
\]

Thus, the general formula for \( f(n) \) is:

\[
f(n) = n + 1
\]

Would you like more details or further explanation?

Here are 5 related questions:

1. What is the value of \( f(10) \)?
2. How would the function change if \( f(1) \) was a different value?
3. Can you derive the closed form of another recursive function?
4. What is the sum of \( f(n) \) from \( n = 1 \) to \( n = 5 \)?
5. How would the graph of \( f(n) = n + 1 \) look?

**Tip**: Recursive definitions are often simplified by recognizing patterns or converting them into a closed formula!

Discover how to solve the recursive function f(1) = 2 and f(n) = f(n−1) + 1 with a step-by-step explanation. Learn the general formula f(n) = n + 1 and understand how recursion leads to an arithmetic sequence. This concept is perfect for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation