# 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, 4), (2, 8), (3, 16), (4, 32) Part A: Is this data modeling an arithmetic sequence or a geometric

###### Question:

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, 4), (2, 8), (3, 16), (4, 32)

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 8th station. Show your work. (4 points

## Answers

Aurora's plan is an illustration of a geometric sequence.The recursive function is [tex]T_n = 2 T_{n-1[/tex].She will complete the 8th station in 512 minutesFrom the question, we have:[tex](x,y) = \{(1,4),(2,8),(3,16),(4,32)\}[/tex](a) The pattern modelTo check if it represents an arithmetic sequence, we calculate the common difference between the y-coordinates[tex]d = 8 - 4 = 4[/tex][tex]d = 16 - 8 = 8[/tex][tex]d = 32 - 16 = 16[/tex]The calculated differences are not the same;This means that, the model is not arithmeticTo check if it represents a geometric sequence, we calculate the ratio between the y-coordinates[tex]r = \frac 84 = 2[/tex][tex]r = \frac{16}8 = 2[/tex][tex]r = \frac{32}{16} = 2[/tex]The calculated ratios are the same;This means that, the model is geometric(b) The recursive functionThe sequence is given as: [tex](x,y) = \{(1,4),(2,8),(3,16),(4,32)\}[/tex]Where:[tex]T_1 = 4[/tex][tex]T_2 = 8[/tex][tex]T_3 = 16[/tex][tex]T_4 = 32[/tex]Rewrite as:[tex]T_1 = 4[/tex][tex]T_2 = 2 \times 4[/tex][tex]T_3 = 2 \times 8[/tex][tex]T_4 = 2 \times 16[/tex]Substitute [tex]T_3 = 16[/tex] in [tex]T_4 = 2 \times 16[/tex][tex]T_4 = 2 \times T_3[/tex]Express 3 as 4 - 1[tex]T_4 = 2 \times T_{4-1[/tex]Represent 4 as n[tex]T_n = 2 \times T_{n-1[/tex]Hence, the recursive sequence is:[tex]T_n = 2 T_{n-1[/tex](c) The explicit formulaIn (a), we have:[tex]r =2[/tex][tex]T_1 = 4[/tex]The nth term of a geometric sequence is:[tex]T_n = T_1 \times r^{n-1}[/tex]So, we have:[tex]T_n = 4\times 2^{n-1}[/tex]Express 4 as [tex]2^2[/tex][tex]T_n = 2^2\times 2^{n-1}[/tex]Apply law of indices[tex]T_n = 2^{2+ n-1}[/tex][tex]T_n = 2^{2-1+ n}[/tex][tex]T_n = 2^{1+ n}[/tex]So, the time to complete the 8th station is:[tex]T_8 = 2^{1+ 8}[/tex][tex]T_8 = 2^{9[/tex][tex]T_8 = 512[/tex]Hence, she will complete the 8th station in 512 minutesRead more about geometric sequence at:https://brainly.com/question/11266123

For those who don't understand:Part A: Geometric as each number is multiplied by 2 We can see it --------------------------------------------------------------------------------------------------------------- Part B: We see that if s is station and t is time then Ts=3 x 2 to the power of 8 minus 1 end exponent Which is the required recursive formula to determine the time =t5 =3 x 2 to the power of 5 minus 1 end exponent =3 x 16 = 48 So on the fifth time is 48 --------------------------------------------------------------------------------------------------------------- Part C: the time she will complete station 8: =t8 =3x2 to the power of 8 minus 1 end exponent =3x2 to the power of 7 =384 the time she will complete station 9: =t9 =3x2 to the power of 9 minus 1 end exponent =3 x 2 to the power of 8 =768 the required time she will complete the 9th station: =t subscript 9 minus t subscript 8 =768-384 =384Remember to change or else u get a 0%