Let's analyze both sequences to determine their general terms $a_n$.

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: $a_n = \frac{n + 2}{(n + 3)^2}$

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: $a_n = \frac{1}{2n}$

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

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?

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