Question

Please i need help right now 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 9th station. Show your work.

Answer 1

Part A:

This data appears to be modeling a geometric sequence because the time to complete each station is increasing by a constant factor. For example, the time to complete station 2 is 4 minutes, which is double the time it took to complete station 1. Similarly, the time to complete station 3 is 8 minutes, which is double the time it took to complete station 2, and so on.

Part B:

To find the time Aurora will take to complete station 5 using a recursive formula, we can use the formula:

a_n = a_1 * r^(n-1)

where a_n is the time it takes to complete station n, a_1 is the time it takes to complete station 1, and r is the common ratio (the factor by which the time increases from one station to the next).

We are given that a_1 = 2 and a_2 = 4, so we can solve for r:

r = a_2 / a_1

= 4 / 2

= 2

Then we can use this value of r to find the time it takes to complete station 5:

a_5 = a_1 * r^(5-1)

= 2 * 2^4

= 2 * 16

= 32

Therefore, Aurora will take 32 minutes to complete station 5.

Part C:

To find the time Aurora will take to complete station 9 using an explicit formula, we can use the formula:

a_n = a_1 * r^(n-1)

where a_n is the time it takes to complete station n, a_1 is the time it takes to complete station 1, and r is the common ratio (the factor by which the time increases from one station to the next).

We are given that a_1 = 2 and a_2 = 4, so we can solve for r:

r = a_2 / a_1

= 4 / 2

= 2

Then we can use this value of r to find the time it takes to complete station 9:

a_9 = a_1 * r^(9-1)

= 2 * 2^8

= 2 * 256

= 512

Therefore, Aurora will take 512 minutes to complete station 9.

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