To find the 5th term of an arithmetic sequence, we first need to identify the common difference (the amount that is added to each term to get the next term) and then use the formula for the $n$-th term of an arithmetic sequence.

Step-by-Step Explanation

Given Sequence:
$\frac{1}{6}, \frac{1}{3}, \frac{1}{2}, \ldots$

1. Identify the first term ($a$):
   The first term, $a$, is $\frac{1}{6}$.

2. Find the common difference ($d$):
   To find the common difference, subtract the first term from the second term, and so on: $d = \frac{1}{3} - \frac{1}{6}$ Let's calculate this:

   $d = \frac{2}{6} - \frac{1}{6} = \frac{1}{6}$

   So, the common difference, $d$, is $\frac{1}{6}$.

3. Use the formula for the $n$-th term of an arithmetic sequence:
   The formula to find the $n$-th term ($a_n$) of an arithmetic sequence is: $a_n = a + (n-1) \cdot d$ Where:

   • $a$ is the first term.
   • $d$ is the common difference.
   • $n$ is the term number.

4. Find the 5th term ($a_5$):
   Substitute $a = \frac{1}{6}$, $d = \frac{1}{6}$, and $n = 5$ into the formula: $a_5 = \frac{1}{6} + (5-1) \cdot \frac{1}{6}$

   Simplify inside the parentheses: $a_5 = \frac{1}{6} + 4 \cdot \frac{1}{6}$

   Multiply: $a_5 = \frac{1}{6} + \frac{4}{6}$

   Combine the fractions: $a_5 = \frac{5}{6}$

Conclusion

The 5th term of the arithmetic sequence $\frac{1}{6}, \frac{1}{3}, \frac{1}{2}, \ldots$ is $\frac{5}{6}$.

Do you want more details, or do you have any questions?

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Related Questions:

1. How do you find the common difference of an arithmetic sequence with more complicated fractions?
2. What is the general formula for the $n$-th term in an arithmetic sequence?
3. How do you find the sum of the first $n$ terms of an arithmetic sequence?
4. How do you determine if a sequence is arithmetic or not?
5. What happens if the common difference is negative?

Tip: Always double-check your common difference to ensure it's correct, as this determines the whole sequence!