Question

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 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

Answer 1

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

=384

Remember to change or else u get a 0%

Answer 2

Aurora's plan is an illustration of a geometric sequence.

• The recursive function is $T_n = 2 T_{n-1$.
• She will complete the 8th station in 512 minutes

From the question, we have:

$(x,y) = \{(1,4),(2,8),(3,16),(4,32)\}$

(a) The pattern model

To check if it represents an arithmetic sequence, we calculate the common difference between the y-coordinates

$d = 8 - 4 = 4$

$d = 16 - 8 = 8$

$d = 32 - 16 = 16$

The calculated differences are not the same;

This means that, the model is not arithmetic

To check if it represents a geometric sequence, we calculate the ratio between the y-coordinates

$r = \frac 84 = 2$

$r = \frac{16}8 = 2$

$r = \frac{32}{16} = 2$

The calculated ratios are the same;

This means that, the model is geometric

(b) The recursive function

The sequence is given as: $(x,y) = \{(1,4),(2,8),(3,16),(4,32)\}$

Where:

$T_1 = 4$

$T_2 = 8$

$T_3 = 16$

$T_4 = 32$

Rewrite as:

$T_1 = 4$

$T_2 = 2 \times 4$

$T_3 = 2 \times 8$

$T_4 = 2 \times 16$

Substitute $T_3 = 16$ in $T_4 = 2 \times 16$

$T_4 = 2 \times T_3$

Express 3 as 4 - 1

$T_4 = 2 \times T_{4-1$

Represent 4 as n

$T_n = 2 \times T_{n-1$

Hence, the recursive sequence is:

$T_n = 2 T_{n-1$

(c) The explicit formula

In (a), we have:

$r =2$

$T_1 = 4$

The nth term of a geometric sequence is:

$T_n = T_1 \times r^{n-1}$

So, we have:

$T_n = 4\times 2^{n-1}$

Express 4 as $2^2$

$T_n = 2^2\times 2^{n-1}$

Apply law of indices

$T_n = 2^{2+ n-1}$

$T_n = 2^{2-1+ n}$

$T_n = 2^{1+ n}$

So, the time to complete the 8th station is:

$T_8 = 2^{1+ 8}$

$T_8 = 2^{9$

$T_8 = 512$

Hence, she will complete the 8th station in 512 minutes

Read more about geometric sequence at:

brainly.com/question/11266123