To find an equation that describes the sequence \(35, 105, 315, \ldots\), let's first look at the ratio of consecutive terms:
- \( \frac{105}{35} = 3 \)
- \( \frac{315}{105} = 3 \)
This indicates that the sequence is geometric, with a common ratio of 3.
Next, we can express the \(n\)-th term of a geometric sequence with the formula:
\[ a_n = a_1 \cdot r^{(n-1)} \]
where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
In this particular sequence:
- The first term \(a_1 = 35\)
- The common ratio \(r = 3\)
Thus, we can write the equation for the \(n\)-th term of the sequence as:
\[ a_n = 35 \cdot 3^{(n-1)} \]
So, the final answer to write in the specified format is:
\[ a_n = 35 \cdot 3^{n-1} \]