Aurora is planning to participate in an event at her school's field day requires her to complete tasks various stations in the fastest time possible.

Step 1
We have, Aurora is practicing and keeping track of her time to complete each station. The x-coordinate is the station number, and the y-coordinate is the time in minutes since the start of the race that she completed the task. ${\displaystyle (1,3),(2,6),(3,12),(4,24).}$
(A) Here, y-coordinate have values as
${\displaystyle a_1=3,a_2=6,a_3=12,a_4=24,a_5\cdots a_n}$
In the above sequence,
${\displaystyle \frac{a_2}{a_1}=\frac{6}{3}=2}$
${\displaystyle \frac{a_3}{a_2}=\frac{12}{6}=2}$
Hence, above sequence data is modeling a geometric sequence because it follow the geometric sequence pattern having same common ratio.
Step 2
(B) We know that for a geometric sequence, ${\displaystyle n^{th}}$ term of sequence having first term a and common ratio r is given as
1) ${\displaystyle a_n=a{r}^{n-1}}$
From the given data, x-coordinate corresponds to station number and y-coordinate corresponds to time t.
For station5, we need to calculate ${\displaystyle 5^{th}}$ term of the sequence given by equation (1).
On substituting ${\displaystyle a=3,r=2,}$ and ${\displaystyle n=5}$ in equation (1) and using explicit formula to determine the time she will complete station 5 as
${\displaystyle a_5=\left(3\right)\cdot {\left(2\right)}^{5-1}}$
${\displaystyle =\left(3\right)\cdot {\left(2\right)}^4}$
${\displaystyle =\left(3\right)\cdot \left(16\right)}$
${\displaystyle =48}$ minutes
Hence, the time she will complete station 5 is 48 minutes.
(C) For station 9, we need to calculate ${\displaystyle 9^{th}}$ term of the sequence given by equation (1).
On substituting ${\displaystyle a=3,r=2,}$ and ${\displaystyle n=5}$ in equation (1) and using explicit formula to determine the time she will complete station 9 as
${\displaystyle a_9=\left(3\right)\cdot {\left(2\right)}^{9-1}}$
${\displaystyle =\left(3\right)\cdot {\left(2\right)}^2}$
${\displaystyle =\left(3\right)\cdot \left(256\right)}$
${\displaystyle =768}$ minutes
Hence, the time she will complete station 9 is 768 minutes.