The given sequence is:

$1, \frac{1}{2}, 2, \frac{1}{3}, 3, \frac{1}{4}, 4, \dots$

Let’s first separate the sequence into two alternating parts:

• Odd positions: $1, 2, 3, 4, \dots$
• Even positions: $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$

Odd-indexed terms:

The terms at odd indices are integers that increase by 1:

$1, 2, 3, 4, \dots$

This can be represented by:

$a_{2n-1} = n$

where $n$ is a positive integer.

Even-indexed terms:

The terms at even indices are fractions of the form:

$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$

This can be represented by:

$a_{2n} = \frac{1}{n+1}$

where $n$ is a positive integer.

General term:

To combine the odd and even terms into a single expression for the general term $a_n$, we use a piecewise function based on whether $n$ is odd or even:

\begin{cases} \frac{1}{\frac{n+1}{2}} & \text{if } n \text{ is even} \\ \frac{n+1}{2} & \text{if } n \text{ is odd} \end{cases}$$ This is the general form for the sequence. Would you like further details or have any questions? --- **Related Questions:** 1. How can you find the sum of the first $$n$$ terms of this sequence? 2. How does the sequence behave as $$n$$ approaches infinity? 3. Can you derive a general rule for sequences alternating between integers and fractions? 4. How can we generalize this process for other types of alternating sequences? 5. What happens if we change the fractions in the even positions to a different progression? **Tip:** When working with alternating sequences, always look at each part separately to identify patterns.