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.