Given the geometric sequence: $10, \frac{15}{2}, \frac{45}{8} \ldots$ Find an explicit formula for $a_{n}$. \[ a_{n}= \] Find $a_{13}=$

Step 1 ：Find the common ratio of the geometric sequence by dividing the second term by the first term, or the third term by the second term: \(r = \frac{\frac{15}{2}}{10} = \frac{3}{4}\) or \(r = \frac{\frac{45}{8}}{\frac{15}{2}} = \frac{3}{4}\)

Step 2 ：Use the explicit formula for a geometric sequence: \(a_{n} = a_{1} * r^{(n-1)}\), where \(a_{1}\) is the first term and r is the common ratio

Step 3 ：Substitute the given values into the formula: \(a_{n} = 10 * (\frac{3}{4})^{(n-1)}\)

Step 4 ：Find \(a_{13}\) by substituting n = 13 into the formula: \(a_{13} = 10 * (\frac{3}{4})^{(13-1)} = 10 * (\frac{3}{4})^{12}\)

Step 5 ：Calculate \((\frac{3}{4})^{12}\) first: \((\frac{3}{4})^{12} = \frac{531441}{16777216}\)

Step 6 ：Multiply this by 10 to get \(a_{13}\): \(a_{13} = 10 * \frac{531441}{16777216} = \frac{5314410}{16777216}\)

Step 7 ：\(\boxed{a_{13} = \frac{5314410}{16777216}}\)