https://file.aiquickdraw.com/mathbot/file/1733918953594iqm271z7.jpg

The problem requires finding the first four terms and the 100th term of the sequence defined by:

\[
a_n = (-1)^n + \frac{n}{n+2}.
\]

### Step 1: Calculate the first four terms
We will substitute \( n = 1, 2, 3, \) and \( 4 \) into the formula.

1. **For \( n = 1 \):**
   \[
   a_1 = (-1)^1 + \frac{1}{1+2} = -1 + \frac{1}{3} = -\frac{2}{3}.
   \]

2. **For \( n = 2 \):**
   \[
   a_2 = (-1)^2 + \frac{2}{2+2} = 1 + \frac{2}{4} = 1 + \frac{1}{2} = \frac{3}{2}.
   \]

3. **For \( n = 3 \):**
   \[
   a_3 = (-1)^3 + \frac{3}{3+2} = -1 + \frac{3}{5} = -\frac{5}{5} + \frac{3}{5} = -\frac{2}{5}.
   \]

4. **For \( n = 4 \):**
   \[
   a_4 = (-1)^4 + \frac{4}{4+2} = 1 + \frac{4}{6} = 1 + \frac{2}{3} = \frac{5}{3}.
   \]

### Step 2: Calculate the 100th term
For \( n = 100 \):
\[
a_{100} = (-1)^{100} + \frac{100}{100+2}.
\]
Since \( (-1)^{100} = 1 \):
\[
a_{100} = 1 + \frac{100}{102} = 1 + \frac{50}{51} = \frac{101}{51}.
\]

### Final Answers:
\[
a_1 = -\frac{2}{3}, \quad a_2 = \frac{3}{2}, \quad a_3 = -\frac{2}{5}, \quad a_4 = \frac{5}{3}, \quad a_{100} = \frac{101}{51}.
\]

Let me know if you'd like me to clarify further! Below are some related questions to deepen understanding.

---

### Related Questions:
1. What happens to the sequence as \( n \to \infty \)? Does it converge or diverge?
2. How does the alternating sign \((-1)^n\) affect the behavior of the sequence?
3. What is the general formula for the sum of the first \( n \) terms of the sequence?
4. How would you modify the sequence to remove the alternating sign?
5. What is the limit of the term \(\frac{n}{n+2}\) as \( n \to \infty \)?

---

### Tip:
When working with alternating sequences like this, analyze the even and odd cases separately if needed to better understand the pattern.

Find the first four terms and the 100th term of the sequence whose nth term is given by a_n = (-1)^n + n / (n + 2).

Learn how to compute the first four terms and the 100th term of a sequence defined by a_n = (-1)^n + n / (n + 2). This step-by-step solution explores alternating sequences and fractional simplifications, suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation