O Sequences, Senes, and Probibility Finding a specified term of a geometric sequence given two terms of the... Elizabet For a given geometric sequence, the $8^{\text {th }}$ term, $a_{8}$, is equal to $\frac{31}{625}$, and the $13^{\text {th }}$ term, $a_{13}$, is equal to -155 . Find the value of the $17^{\text {th }}$ term, $a_{17}$. If applicable, write your answer as a fraction. \[ a_{17}=\mathbb{0} \] 믐 X 5

Step 1 ：For a given geometric sequence, the 8th term, \(a_{8}\), is equal to \(\frac{31}{625}\), and the 13th term, \(a_{13}\), is equal to -155 . We are asked to find the value of the 17th term, \(a_{17}\).

Step 2 ：We can use the formula for the nth term of a geometric sequence, which is \(a_n = a \cdot r^{(n-1)}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

Step 3 ：Given that \(a_{8} = \frac{31}{625}\) and \(a_{13} = -155\), we can calculate the common ratio \(r\) as \(-10\).

Step 4 ：Substituting \(a = -\frac{31}{6250000000}\), \(r = -10\), and \(n = 17\) into the formula, we get \(a_{17} = a \cdot r^{(n-1)}\).

Step 5 ：Calculating the above expression, we find that \(a_{17} = -49600000.0\).

Step 6 ：So, the 17th term of the geometric sequence is \(-49600000.0\).

Step 7 ：Final Answer: \(\boxed{-49600000.0}\)