https://file.aiquickdraw.com/mathbot/file/1729315529197bemcmll2.jpg

Let's analyze both sequences to determine their general terms \(a_n\).

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### **1st sequence: \( \frac{3}{16}, \frac{4}{25}, \frac{5}{36}, \frac{6}{49}, \ldots \)**

Observe that:
- The numerators are: \(3, 4, 5, 6, \ldots\), which is \(n + 2\) for \(n = 1, 2, 3, \ldots\).
- The denominators are: \(16, 25, 36, 49, \ldots\), which correspond to \(4^2, 5^2, 6^2, 7^2, \ldots\) or \((n + 3)^2\).

Thus, the general term for this sequence is:
\[
a_n = \frac{n + 2}{(n + 3)^2}
\]

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### **2nd sequence: \( \frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \ldots \)**

Here:
- The numerators are all \(1\).
- The denominators are \(2, 4, 6, 8, \ldots\), which correspond to \(2n\) for \(n = 1, 2, 3, \ldots\).

Thus, the general term for this sequence is:
\[
a_n = \frac{1}{2n}
\]

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If anything is unclear, or if you have questions, feel free to ask!

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### **Related Questions:**
1. How can you determine if a sequence is arithmetic or geometric?
2. What are the differences between converging and diverging sequences?
3. Can the general term \(a_n\) involve factorials?
4. How do we find the sum of an infinite sequence if it converges?
5. What is the difference between sequences and series?

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**Tip:** Identifying patterns in numerators and denominators separately can simplify finding the general formula for complex sequences.

For each sequence, find a formula for the general term, a_n. Sequences start with n = 1. For example, answer n^2 if given the sequence: 1, 4, 9, 16, 25, 36, ...

Learn how to determine the general term a_n for sequences, with examples including fractions and pattern recognition. Suitable for students in Grades 9-12.

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