College Algebra, Chapter 9, 9.3, Section 9.3, Problem 34

Find the common ratio, the fifth term and the $n$th term of the geometric sequence $\displaystyle t, \frac{t^2}{2}, \frac{t^3}{4}, \frac{t^4}{8},...$

To find a formula for the $n$th term of this sequence, we need to find $a$ and $r$. Clearly, $a = t$. To find $r$, we find the ratio of any two consecutive terms.

For instance, $\displaystyle r = \frac{\displaystyle \frac{t^3}{4}}{\displaystyle \frac{t^2}{2}} = \frac{t}{2}$, thus, the $n$th term is

$\displaystyle a_n = t \left( \frac{t}{2} \right)^{n-1}$

So the fifth term is

$\displaystyle a_5 = t \left( \frac{t}{2} \right)^{5-1} = t \left( \frac{t}{2} \right)^4 = \frac{t^5}{16}$