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

Step-by-step explanation:

an arithmetic sequence is a sequence of terms, where every term is created by adding a certain constant to the previous term.

a geometric sequence is when every term is created by multiplying the previous term by a certain constant.

the x coordinate serves as "n" counter of the terms in the sequence.

and the y coordinate is the actual sequence term of that "n".

so, we have

a1 = 4

a2 = 8

a3 = 16

a4 = 32

A)

I think, now it is obvious : the sequence is built by multiplying the previous term by 2.

and this is therefore a geometric sequence.

B)

a2 = a1 × 2 = 4×2 = 8 minutes

a3 = a2 × 2 = a1 × 2 × 2 = 4 × 2 × 2 = 4 × 2² = 16 minutes

a4 = a3 × 2 = 16 × 2 = 32 minutes

a5 = a4 × 2 = 32 × 2 = 64 minutes

C)

based on B we see

an = an-1 × 2 = a1 × 2^(n-1) = 4 × 2^(n-1) = 2² × 2^(n-1) =

= 2^(n+1) minutes

therefore

a8 = 2^(n+1) = 2⁹ = 512 minutes