Aurora is planning to participate in an event at her school's field day that requires her to complete tasks at various stations in the fastest time
possible. to prepare for the event, she 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.
(1,3), (2, 6), (3, 12), (4,24)
part a: is this data modeling an algebraic sequence or a geometric sequence? explain your answer. (2 points)
part b: use a recursive formula to determine the time she will complete station 5. show your work. (4 points)
part c: use an explicit formula to find the time she will complete the 9th station. show your work. (4 points) here's a photo just in case

answer is included in the explanation.

step-by-step explanation:

part a:  

let’s look the y-coordinates.  

this is the pattern we need to look at to determine if we have an arithmetic or geometric sequence.  

the y’s in order (of the x’s) are 3,6,12,24 .  

arithmetic sequences have a common difference (term mine previous term is the same). if it arithmetic then the following would be true:  

24-12 = 12-6 = 6-3  

geometric sequences have a common ratio (term divided by previous is the same). if is geometric then the following would be true:  

24/12 = 12/6 = 6/3  

let’s check for which is true:  

i’m going to simplify each piece in the equalities here:  

24-12 = 12-6 = 6-3  

12 = 6 = 3  

this is definitely not true. we do not have the same number on any side of an equal sign. that is, 3 is not 6, 6 is not 12, and 12 is not 3. they would all have to be the same number in order for us to say 3,6,12,24 is arithmetic.  

let’s check:  

24/12 = 12/6 = 6/3  

2 = 2 = 2  

this is true. this means the sequence is geometric with a common ratio of 2.  

part b:  

recursive formula is a formula that uses previous term(s) in a sequence to define a given term in the sequence.  

there is a formula for geometric sequences for this. we actually already mentioned it:  

let’s put this into symbols:  

we already determined that r is 2.  

when writing recursive forms, it is important to state one point that you know in that sequence. people usually just go with the first term.  

so the recursive definition is with . was given to us in the point (1,3).  

now people don’t normally like the division by the previous term, so they multiply both sides by the previous term to obtain this recursive form instead:  

with  

so we want to use to determine what happens at x=5 (n=5) since x represents station number.  

let’s plug in 5 for n in our final recursive form:  

with given  

we know since we have the point (4,24) in our list.  

this says at station 5 that is has been 48 minutes since the beginning of the race for aurora.  

part c:  

there is a formula for explicit form for a geometric sequence.  

it is where is the first term and is the common ratio.  

we have (1,3) which means .  

we also found the common ratio earlier by dividing term by previous term giving us r=2.  

this means the explicit form for our sequence is: .  

so we want the time it will take her to complete the 9th station (x=9 or n=9 in this case).  

plug in 9 for n:  

with  

this says it takes 768 minutes for her to complete the 9th station.  

true

step-by-step explanation: