To find the [tex]\( n \)[/tex]-th term of the given sequence, observe the pattern in the terms provided:
[tex]\[
\frac{1}{3}, \quad \frac{1}{6}, \quad \frac{1}{9}, \quad \frac{1}{12}, \quad \ldots
\][/tex]
1. Identify the Pattern:
Each term has the numerator 1. We need to examine the denominators to find a pattern:
[tex]\[
3, \quad 6, \quad 9, \quad 12, \quad \ldots
\][/tex]
These denominators form an arithmetic sequence where each term increases by 3.
2. Form the Denominator:
Notice that the first term corresponds to the denominator [tex]\(3\)[/tex] (which is [tex]\(3 \times 1\)[/tex]), the second term corresponds to the denominator [tex]\(6\)[/tex] (which is [tex]\(3 \times 2\)[/tex]), and so forth. In general, the denominator of the [tex]\( n \)[/tex]-th term is [tex]\( 3n \)[/tex].
3. Write the [tex]\( n \)[/tex]-th Term:
Since every term in the sequence has the numerator 1, and the denominator for the [tex]\( n \)[/tex]-th term is [tex]\( 3n \)[/tex], we can write the [tex]\( n \)[/tex]-th term as:
[tex]\[
\frac{1}{3n}
\][/tex]
The general expression for the [tex]\( n \)[/tex]-th term of the sequence is:
[tex]\[
\boxed{\frac{1}{3n}}
\][/tex]
Now, let's illustrate this by determining the first few terms in the sequence to verify our formula:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[
\frac{1}{3 \times 1} = \frac{1}{3}
\][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[
\frac{1}{3 \times 2} = \frac{1}{6}
\][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[
\frac{1}{3 \times 3} = \frac{1}{9}
\][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[
\frac{1}{3 \times 4} = \frac{1}{12}
\][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[
\frac{1}{3 \times 5} = \frac{1}{15}
\][/tex]
These computations align with the given sequence, validating that our expression [tex]\(\frac{1}{3n}\)[/tex] for the [tex]\( n \)[/tex]-th term is correct.
