The terms in a geometric sequence are found by repeatedly multiplying the last term by a constant, called the common ratio.
a)
To find the common ratio, pick a term of the sequence and divide it by the previous term.
We can calculate the common ratio using a table:
| $n$ |
$1$ |
$2$ |
$3$ |
$4$ |
| $a_n$ |
$\var{a*r}$ |
$\var{a*r^2}$ |
$\var{a*r^3}$ |
$\var{a*r^4}$ |
| Common ratio |
|
$\displaystyle\frac{\var{a*r^2}}{(\var{a*r})} = \var{r}$ |
$\displaystyle\frac{\var{a*r^3}}{\var{a*r^2}} = \var{r}$ |
$\displaystyle\frac{\var{a*r^4}}{(\var{a*r^3})} = \var{r}$ |
The common ratio is $\var{d}$.
b)
The general formula for the $n^\text{th}$ term of a geometric sequence is
\[\displaystyle {a_n=ar^{(n-1)}\text{,}}\]
where $a$ is the first term, and $r$ is the common ratio.
So the formula for this sequence is
\[
\begin{align}
a_n&=ar^{(n-1)}\\
&=\var{a*r}\times(\var{r})^{(n-1)}\\
&=(\var{a} \times (\var{r}))(\var{r})^{n-1}\\
&=\var{a}(\var{r})^n\text{.}
\end{align}
\]
c)
We know from part b) that the $n^{th}$ term for this sequence is $a_n = \var{a}(\var{r})^n$.
Therefore the $\var{nth}^{th}$ term in the sequence is
\[
\begin{align}
a_\var{nth} &= \var{a}(\var{r})^\var{nth}\\
&= \var{a} \times (\var{{r}^{nth}})\\
&= \var{{a}*{r}^{nth}}.
\end{align}
\]