https://r2.erweima.ai/mathbot/image/1725800813344qsr2z6ji.jpg

It seems like you've uploaded an image containing some math problems. Let me extract and analyze the content to help you with the questions. Please hold on for a moment.The image contains several mathematical problems, mostly involving sequences and series. Here's a summary of the problems identified in the image:

1. **Problem 1:**
   Prove that the sequence defined by \(u_n = \frac{3n + 1}{4n + 5}\) is convergent and find its limit.

2. **Problem 2:**
   Determine whether the series \(\sum_{n=1}^{\infty} \frac{1}{n(n+2)}\) converges, and if so, find its sum.

3. **Problem 3:**
   Prove by induction that \(S_n = \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{n(n+1)} = 1 - \frac{1}{n+1}\).

4. **Problem 4:**
   Solve the system of linear equations:
   \[
   \begin{cases}
   x + y + z = 6 \\
   2x - y + 3z = 14 \\
   -x + 4y + 2z = 8
   \end{cases}
   \]

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### 5 Questions Related to the Image Content:

1. What is the limit of a sequence and how is it determined?
2. How do you test a series for convergence?
3. What are the steps involved in proving a statement using mathematical induction?
4. What methods can be used to solve a system of linear equations?
5. How can partial fraction decomposition be used in series?

**Tip:** When solving sequences and series problems, always consider applying known tests (like the comparison test, ratio test, or root test) to determine convergence.

The image contains several mathematical problems, mostly involving sequences and series, including proving convergence, testing series for convergence, proving by induction, and solving a system of linear equations.

This page covers advanced mathematical problems including proving the convergence of sequences, testing series for convergence, proving by mathematical induction, and solving systems of linear equations. Suitable for high school students in Grades 11-12.

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