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, 3), (2, 6), (3, 12), (4, 24) Use a recursive formula to determine the time she will complete station 5. Show your work.

To determine the time she will complete station 5 using a recursive formula, let's analyze the pattern in the given data points and then derive a recursive formula.

### Step 1: Analyze the Pattern
The given data points are:
- Station 1: \( t_1 = 3 \) minutes
- Station 2: \( t_2 = 6 \) minutes
- Station 3: \( t_3 = 12 \) minutes
- Station 4: \( t_4 = 24 \) minutes

We need to find a pattern between consecutive times.

### Step 2: Identify the Pattern
Observe how the times change from one station to the next:
- \( t_2 = 2 \times t_1 \) 
- \( t_3 = 2 \times t_2 \) 
- \( t_4 = 2 \times t_3 \) 

This pattern shows that the time to complete each station doubles the time taken at the previous station.

### Step 3: Recursive Formula
The recursive formula based on this pattern can be written as:
\[
t_n = 2 \times t_{n-1}
\]
where \( t_n \) is the time at station \( n \), and \( t_{n-1} \) is the time at the previous station.

### Step 4: Calculate \( t_5 \)
Now, use the recursive formula to calculate the time at station 5:
\[
t_5 = 2 \times t_4
\]
Given \( t_4 = 24 \) minutes:
\[
t_5 = 2 \times 24 = 48 \text{ minutes}
\]

### Conclusion
The time she will complete station 5 is **48 minutes**.

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. What if the time increment wasn't doubling but another pattern?
2. How would the recursive formula change if the time difference increased by a constant amount instead of multiplying?
3. Can you find a non-recursive (explicit) formula for this sequence?
4. How would the sequence change if an initial time, \( t_1 \), were different?
5. What is the total time taken to complete all 5 stations?

**Tip:** Recursive formulas are particularly useful for sequences where each term depends directly on the previous one. Always look for a pattern in how each term changes from the previous one.

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, 3), (2, 6), (3, 12), (4, 24) Use a recursive formula to determine the time she will complete station 5. Show your work.

Learn how to use a recursive formula to predict the time to complete station 5 in a sequence where time doubles at each station. Understand the pattern in the sequence with detailed step-by-step analysis. Suitable for students in Grades 9-11.

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