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madison6592
17.03.2020 •
Mathematics
What’s the answer to this question becaus I am stuck
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Ответ:
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:
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:
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:
this says it takes 768 minutes for her to complete the 9th station.