https://file.aiquickdraw.com/mathbot/file/172948561737779wagm07.jpg

The image you uploaded contains several sequences. Let me help break them down one by one:

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**f.**  
\[
\sqrt{5}, \ 1 + \sqrt{5}, \ 2 + \sqrt{5}, \ 3 + \sqrt{5}, \ldots
\]  
This sequence appears to be an arithmetic progression where the common difference is 1, and each term is added to \(\sqrt{5}\). The general form of the \(n\)-th term can be written as:
\[
T_n = n + \sqrt{5}
\]

---

**g.**  
\[
\frac{1}{2}, \ \frac{1}{3}, \ \frac{1}{4}, \ \frac{1}{5}, \ldots
\]  
This sequence is a harmonic progression where the denominator increases by 1. The general term is:
\[
T_n = \frac{1}{n+1}
\]
since the first term is \(\frac{1}{2}\).

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**h.**  
\[
\frac{3}{16}, \ \frac{3}{8}, \ \frac{3}{4}, \ \frac{3}{2}, \ldots
\]  
This is a geometric sequence where the first term is \(\frac{3}{16}\), and the common ratio is 2. The general term of the sequence can be written as:
\[
T_n = \frac{3}{16} \cdot 2^{n-1}
\]

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**i.**  
\[
a, \ ab, \ ab^2, \ ab^3, \ldots
\]  
This is a geometric sequence where the first term is \(a\), and the common ratio is \(b\). The general term of the sequence is:
\[
T_n = a \cdot b^{n-1}
\]

---

**j.**  
\[
a, \ a + k^2, \ a + 2k^2, \ a + 3k^2, \ldots
\]  
This appears to be an arithmetic sequence where the common difference is \(k^2\). The general term can be written as:
\[
T_n = a + (n-1)k^2
\]

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Would you like more details on any of these sequences or have specific questions?

Here are some related questions for further exploration:
1. What is the sum of the first \(n\) terms of a harmonic sequence (as in g)?
2. Can the sequence in h be written in summation form?
3. How does the general form of an arithmetic progression help solve real-world problems?
4. In sequence i, what happens if \(b\) is a fraction or negative number?
5. How would you represent the sequence in j graphically?

**Tip:** In a geometric sequence, multiplying terms by a constant common ratio makes it easier to express the entire sequence using exponents.

Analyze sequences from the given image. These sequences are labeled as f, g, h, i, and j, each following distinct mathematical patterns.

Explore different types of sequences, including arithmetic, harmonic, and geometric sequences. Learn how to derive general terms and formulas for these patterns, with detailed breakdowns of sequences involving square roots, fractions, and exponents. Suitable for Grades 9-12.

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