For a glven geometric sequence, the 8^th term, a_(8) Is equal to (31)/(27) and the 12^th term, a_(12) Is equal to 93 Find the value of the 16^th term, a_(16) If applicat write your answer as a fraction. a_(16)=square

To find the value of the 16th term, $a_{16}$, we can use the formula for the nth term of a geometric sequence:<br /><br />$a_n = a_1 \cdot r^{(n-1)}$<br /><br />where $a_n$ is the nth term, $a_1$ is the first term, and $r$ is the common ratio.<br /><br />Given that the 8th term, $a_8$, is equal to $\frac{31}{27}$ and the 12th term, $a_{12}$, is equal to 93, we can set up two equations:<br /><br />$a_8 = a_1 \cdot r^7 = \frac{31}{27}$<br /><br />$a_{12} = a_1 \cdot r^{11} = 93$<br /><br />We can solve these equations simultaneously to find the values of $a_1$ and $r$.<br /><br />Dividing the second equation by the first equation, we get:<br /><br />$\frac{a_{12}}{a_8} = \frac{a_1 \cdot r^{11}}{a_1 \cdot r^7} = r^4$<br /><br />Substituting the given values, we have:<br /><br />$\frac{93}{\frac{31}{27}} = r^4$<br /><br />Simplifying the left side, we get:<br /><br />$\frac{93 \cdot 27}{31} = r^4$<br /><br />Calculating the left side, we have:<br /><br />$81 = r^4$<br /><br />Taking the fourth root of both sides, we get:<br /><br />$r = 3$<br /><br />Substituting this value of $r$ back into the first equation, we can solve for $a_1$:<br /><br />$a_1 \cdot 3^7 = \frac{31}{27}$<br /><br />Simplifying the left side, we have:<br /><br />$a_1 \cdot 2187 = \frac{31}{27}$<br /><br />Dividing both sides by 2187, we get:<br /><br />$a_1 = \frac{31}{27 \cdot 2187}$<br /><br />Simplifying the right side, we have:<br /><br />$a_1 = \frac{31}{59049}$<br /><br />Now that we have found the values of $a_1$ and $r$, we can find the value of the 16th term, $a_{16}$:<br /><br />$a_{16} = a_1 \cdot r^{15}$<br /><br />Substituting the values of $a_1$ and $r$, we have:<br /><br />$a_{16} = \frac{31}{59049} \cdot 3^{15}$<br /><br />Simplifying the right side, we get:<br /><br />$a_{16} = \frac{31 \cdot 14348907}{59049}$<br /><br />Therefore, the value of the 16th term, $a_{16}$, is $\frac{31 \cdot 14348907}{59049}$.