To determine the general term $a_n$ for the sequence provided, let’s analyze the given terms.

The sequence is:

$\left\{ \frac{6}{2}, \frac{6}{4}, \frac{6}{8}, \frac{6}{16}, \frac{6}{32}, \dots \right\}$

Step 1: Identify the Denominator Pattern

• The denominators are $2, 4, 8, 16, 32, \dots$.
• This sequence represents powers of $2$, specifically $2^1, 2^2, 2^3, 2^4, 2^5, \dots$.

Thus, the denominator of the $n$-th term is $2^n$.

Step 2: General Term

The numerator in all terms is constant: $6$.

Hence, the general term $a_n$ can be written as:

$a_n = \frac{6}{2^n}$

Final Answer:

$a_n = \frac{6}{2^n}$

Let me know if you'd like a detailed explanation or have any questions!

Related Questions:

1. What is the general term of a sequence with powers of $3$ in the denominator?
2. How do you find the sum of a geometric sequence with a given $a_n$?
3. What happens to $a_n$ as $n \to \infty$ for $a_n = \frac{6}{2^n}$?
4. How can you derive a formula for an arithmetic sequence?
5. How do you identify patterns in sequences with constant numerators?

Tip:

To find the general term in sequences, look for patterns in numerators and denominators separately (e.g., constant values, powers, or arithmetic progressions).