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 po

iohanetc 2021-02-25 Answered
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.
Part B: Use a recursive formula to determine the time she will complete station 5.
Part C: Use an explicit formula to find the time she will complete the 9th station.
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Expert Answer

stuth1
Answered 2021-02-26 Author has 97 answers

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. (1,3),(2,6),(3,12),(4,24).
(A) Here, y-coordinate have values as
a1=3,a2=6,a3=12,a4=24,a5,an
In the above sequence,
a2a1=63=2
a3a4=126=2
Hence, above sequence data is modeling a geometric sequence because it follow the geometric sequence pattern having same common ratio.
(B) We know that for a geometric sequence, nth term of sequence having first term a and common ratio r is given as an=arn1 ⋯⋯(1)
From the given data, x-coordinate corresponds to station number and y-coordinate corresponds to time t.
For station5, we need to calculate 5-th term of the sequence given by equation (1).
On substituting a=3,r=2,andn=5 in equation (1) and using explicit formula to determine the time she will complete station 5 as
a5=(3)(2)51
=(3)(2)4
=(3)(16)
=48 minutes
Hence, the time she will complete station 5 is 48 minutes.
(C)
For station 9, we need to calculate 9th term of the sequence given by equation (1).
On substituting a=3,r=2,andn=5 in equation (1) and using explicit formula to determine the time she will complete station 9 as
a9=(3)(2)91
=(3)(2)8
=(3)(256)
=768 minutes
Hence, the time she will complete station 9 is 768 minutes.

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