Describe the pattern, write the next term, and write a rule for the \( n \)th term of the sequens \( \frac{2}{3}, \frac{2}{6}, \frac{2}{9}, \frac{2}{12}, \ldots \) The pattern is The next term is The \( n \)th term is \( a_{n}= \)

**Step 1: Identify the pattern** The sequence is given by \[ \frac{2}{3}, \quad \frac{2}{6}, \quad \frac{2}{9}, \quad \frac{2}{12}, \quad \ldots \] The numerator is always \(2\). The denominator starts at \(3\) and increases by \(3\) each time. This means that the denominator is a multiple of \(3\). **Step 2: Write the next term** Since the last term provided is \(\frac{2}{12}\) and the denominator increases by \(3\), the next denominator is \[ 12 + 3 = 15. \] Thus, the next term is \[ \frac{2}{15}. \] **Step 3: Write the rule for the \(n\)th term** In the sequence, the denominator for the \(n\)th term is \(3n\) (since for \(n=1\) we have \(3\), for \(n=2\) we have \(6\), and so on), and the numerator is constant \(2\). Hence, the rule for the \(n\)th term is \[ a_{n} = \frac{2}{3n}. \] **Summary** The pattern is: the numerator is constant at \(2\) and the denominator increases by \(3\) with each term. The next term is: \(\frac{2}{15}\). The \(n\)th term is: \[ a_{n} = \frac{2}{3n}. \]