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, 2), (2, 4), (3, 8), (4, 16)Part A: Is this data modeling an arithmetic 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 10th station. Show your work. (4 points)

Question

Aurora is planning to participate in an event at her school&#039;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, 2), (2, 4), (3, 8), (4, 16)Part A: Is this data modeling an arithmetic 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 10th station. Show your work. (4 points)

madeleinejames20 · Accepted Answer

Part A: To determine whether the given data is modeling an arithmetic sequence or a geometric sequence, we need to examine the pattern in the y-coordinates (time values).An arithmetic sequence is a sequence in which the difference between consecutive terms is constant. In other words, if we subtract any term from the one that follows it, we will obtain the same value.A geometric sequence, on the other hand, is a sequence in which the ratio between consecutive terms is constant. If we divide any term by the one that precedes it, we will obtain the same value.Let&#039;s examine the given data: (1, 2), (2, 4), (3, 8), (4, 16)To determine if it is an arithmetic sequence, we will check if the differences between consecutive y-coordinates are constant:4−2=2, 8−4=4, 16−8=8The differences between consecutive terms are not constant; hence, the data does not model an arithmetic sequence.To determine if it is a geometric sequence, we will check if the ratios between consecutive y-coordinates are constant:42=2, 84=2, 168=2The ratios between consecutive terms are constant, specifically 2. Therefore, the given data models a geometric sequence.Part B: Now, let&#039;s use a recursive formula to determine the time Aurora will complete station 5. In a geometric sequence, a recursive formula is of the form:an=an−1·rwhere an represents the nth term, an−1 represents the previous term, and r represents the common ratio.From the given data, we can observe that the common ratio (r) is 2. We can start with the initial term a1=2, and use the recursive formula to find the time for station 5:a5=a4·ra4=16 (as given)a5=16·2=32Therefore, the time it will take Aurora to complete station 5 is 32 minutes.Part C: Next, let&#039;s use an explicit formula to find the time Aurora will complete the 10th station. In a geometric sequence, the explicit formula is of the form:an=a1·rn−1where an represents the nth term, a1 represents the initial term, r represents the common ratio, and n represents the position of the term.Using the explicit formula, we can find the time for the 10th station:a10=a1·r10−1a1=2 (as given)r=2 (as observed from the data)a10=2·210−1=2·29=2·512=1024Therefore, the time it will take Aurora to complete the 10th station is 1024 minutes.

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