This image contains handwritten math problems written in Persian, which primarily focus on sequences and series. Let’s break down the key content:

1. First Problem

The first problem defines sequences:

• $a_n = \frac{n - 1}{n + 2}$
• $b_n = 3^{n+1}$
• $c_n = \left(\frac{1}{2}\right)^{n-1}$

The task seems to require evaluating or analyzing these sequences.

2. Second Problem

You are given the recurrence relation:
$a_{n+1} = a_n + (-1)^n \quad \text{with} \quad a_1 = 1.$
The problem asks to write the first few terms of this sequence.

3. Third Problem

Evaluate the following mathematical expressions: $\frac{1}{4} \quad \text{and} \quad \frac{1}{25}.$

4. Fourth Problem

The task involves determining a relation or formula for sequences involving specific terms like $1, 2, 3, \dots, 9, 27$, and possibly finding a pattern.

5. Fifth Problem

You are asked to analyze the recurrence relation: $a_1 = 2, \quad a_{n+1} = -\frac{1}{a_n}.$
The goal is to compute the terms or behavior of the sequence.

If you'd like detailed solutions or explanations for any of these parts, let me know!

Related Questions:

1. How do you determine the general term of a sequence given a recurrence relation?
2. What are common techniques to analyze geometric and arithmetic sequences?
3. How can you solve a sequence involving alternating signs?
4. What is the behavior of a sequence defined by recursive inversions?
5. How do you prove or simplify sequences defined as fractions?

Tip: To analyze recurrence relations like $a_{n+1} = -\frac{1}{a_n}$, calculate the first few terms to detect periodic behavior or patterns.