Find the general term of the sequence $$ \frac32,\frac{\text{ }6}{5},\frac{9\text{ }}{10},\frac{\text{ }12}{17}, $$$$ \frac{15}{26} $$

To find the general term of the sequence 

\[
\frac{3}{2}, \frac{6}{5}, \frac{9}{10}, \frac{12}{17}, \frac{15}{26},
\]

let's analyze the sequence of numerators and denominators separately.

### Step 1: Identify the pattern in the numerators
The numerators of the sequence are $3, 6, 9, 12, 15$.

This is an arithmetic sequence with the first term $a_1 = 3$ and a common difference $d = 3$.

The general term for an arithmetic sequence is:
\[
\text{Numerator: } a_n = 3n
\]

### Step 2: Identify the pattern in the denominators
The denominators of the sequence are $2, 5, 10, 17, 26$.

Let's examine this pattern. The sequence of denominators can be described as:

- $2 = 1^2 + 1$
- $5 = 2^2 + 1$
- $10 = 3^2 + 1$
- $17 = 4^2 + 1$
- $26 = 5^2 + 1$

So, the denominators are following the sequence $n^2 + 1$.

Thus, the general term for the denominator is:
\[
\text{Denominator: } b_n = n^2 + 1
\]

### Step 3: Combine the numerator and denominator
The general term of the sequence is:
\[
a_n = \frac{3n}{n^2 + 1}
\]

So the general term for the given sequence is:
\[
a_n = \frac{3n}{n^2 + 1}
\]

Would you like to delve into further details or have any questions about the sequence?

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Here are 5 related questions for further exploration:
1. How would the sequence change if the numerator followed a different arithmetic progression?
2. Can the sequence be expressed as a sum of simpler sequences?
3. What is the behavior of the sequence as $n$ approaches infinity?
4. How would the sequence change if the denominator followed a different quadratic pattern?
5. How can you determine if this sequence converges or diverges?

**Tip:** When analyzing sequences, always consider looking at the numerators and denominators separately, especially if the sequence is in fractional form. This can often simplify the process of finding a general term.

Discover how to find the general term of the sequence 3/2, 6/5, 9/10, 12/17, 15/26 using arithmetic sequences and quadratic patterns. Learn the steps to identify the numerator and denominator patterns, and derive the general formula. Suitable for advanced high school students.

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