It looks like you have provided a sequence of numbers: $\frac{1}{2}, 1, \frac{7}{4}, \frac{5}{2}$. Let’s analyze the sequence to identify its pattern.

We can first express all terms with the same denominator to better understand the differences between the terms:

• $\frac{1}{2} = \frac{2}{4}$
• $1 = \frac{4}{4}$
• $\frac{7}{4}$ stays as is.
• $\frac{5}{2} = \frac{10}{4}$

So the sequence becomes: $\frac{2}{4}, \frac{4}{4}, \frac{7}{4}, \frac{10}{4}$.

Now, let’s find the differences between consecutive terms:

• $\frac{4}{4} - \frac{2}{4} = \frac{2}{4} = \frac{1}{2}$
• $\frac{7}{4} - \frac{4}{4} = \frac{3}{4}$
• $\frac{10}{4} - \frac{7}{4} = \frac{3}{4}$

We observe that after the first difference, the remaining differences are constant at $\frac{3}{4}$. This suggests that after the first term, the sequence follows an arithmetic progression with a common difference of $\frac{3}{4}$.

Next Term in the Sequence

To find the next term: $\frac{5}{2} + \frac{3}{4} = \frac{10}{4} + \frac{3}{4} = \frac{13}{4}$ So, the next term is $\frac{13}{4}$.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How do you find the common difference in an arithmetic sequence?
2. What is the general formula for the $n$-th term of an arithmetic sequence?
3. How can we determine if a sequence is arithmetic or geometric?
4. How do you find the sum of the first $n$ terms in an arithmetic sequence?
5. Can an arithmetic sequence have negative terms or a decreasing pattern?

Tip: In arithmetic sequences, once you find a pattern in the differences between consecutive terms, you can predict future terms using the common difference.