To determine if the given sequence $A_n$ is convergent, divergent, or divergent due to being unbounded, we analyze the general term $A_n = \frac{9 + 3n}{4 + 2n}$.

Step 1: Analyze the general term

The sequence $A_n = \frac{9 + 3n}{4 + 2n}$ has a numerator $9 + 3n$ and denominator $4 + 2n$, both linear functions of $n$.

To find the behavior as $n \to \infty$: $A_n = \frac{9 + 3n}{4 + 2n}.$ Divide numerator and denominator by $n$: $A_n = \frac{\frac{9}{n} + 3}{\frac{4}{n} + 2}.$ As $n \to \infty$, $\frac{9}{n} \to 0$ and $\frac{4}{n} \to 0$, so: $A_n \to \frac{3}{2}.$

Step 2: Conclusion

The sequence converges, and its limit is $\frac{3}{2}$.

Final Answer:

• Convergent sequence whose limit is $\frac{3}{2}$.

Would you like a detailed explanation of how I simplified the sequence?

Related Questions:

1. How does dividing by the highest power of $n$ simplify the analysis of sequences?
2. What does it mean for a sequence to be divergent versus unbounded?
3. What are other techniques to find the limit of a rational sequence?
4. How would the conclusion change if the numerator or denominator were quadratic instead of linear?
5. Can you provide another example of a convergent sequence with a different limit?

Tip:

When analyzing limits of sequences, dividing the numerator and denominator by the highest power of $n$ present in the denominator is a helpful strategy to simplify.