Let's address each part of your question step by step.
A. Write the values given in the scenario as a sequence.
The values of Willow's salary for each of the four years can be represented as the following sequence:
- Year 1: $48,000
- Year 2: $50,400
- Year 3: $52,920
- Year 4: $55,566
So, the sequence of her salaries is: \[ 48000, 50400, 52920, 55566 \]
B. Is this an example of an arithmetic sequence, a geometric sequence, or another type of sequence? Explain.
This is an example of a geometric sequence because the salary increases by a constant percentage each year. In a geometric sequence, each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Arithmetic sequences, on the other hand, involve additions/subtractions of a fixed number.
C. Create a function that models this sequence/scenario.
To model this scenario with a function, we can express her salary as a function of time, denoting the number of years since the start. We can identify that her salary increases exponentially due to the constant percentage increase.
Let:
- \( S(n) \) represent Willow's salary in year \( n \),
- \( S(0) = 48000 \) (Her starting salary),
- \( r \) be the common ratio (which we will find in part D).
The general form of the function can be expressed as: \[ S(n) = S(0) \cdot (1 + r)^n \] where \( n \) is the number of years and \( r \) is the percentage increase expressed as a decimal.
D. By what percentage is Willow’s salary increasing each year? Show your work.
To find the percentage increase, we can calculate the ratios of consecutive years:
-
From Year 1 to Year 2: \[ r_1 = \frac{50400 - 48000}{48000} = \frac{2400}{48000} = 0.05 \text{ (which is 5%)} \]
-
From Year 2 to Year 3: \[ r_2 = \frac{52920 - 50400}{50400} = \frac{2520}{50400} = 0.05 \text{ (which is 5%)} \]
-
From Year 3 to Year 4: \[ r_3 = \frac{55566 - 52920}{52920} = \frac{2646}{52920} = 0.05 \text{ (which is 5%)} \]
Thus, Willow’s salary is increasing by 5% each year.
E. Willow wants to use your function to predict her salary in the future. She wants to make sure her predictions are made to a reasonable level of accuracy. How can she do this?
To ensure that her predictions are made to a reasonable level of accuracy, Willow should:
-
Validating the Model: Compare the predictions from the model against actual historical data to confirm that the model holds true. If the actual salaries deviate significantly from the predictions, the model may need adjustments.
-
Regular Updates: Regularly review and update the model as her salary structure may change due to factors such as promotions, changes in company policy, or economic conditions.
-
Consider Additional Variables: If other factors come into play (e.g., bonuses, inflation rates, etc.), Willow should expand her model to account for those factors.
-
Make Predictions Over Shorter Periods: Predictions should focus on a few years ahead rather than long-term forecasts, as uncertainty tends to increase over time.
-
Sensitivity Analysis: Assess how changes in the percentage increase could affect predictions, helping to understand the potential range of outcomes.
By using a combination of these methods, Willow can improve the accuracy of her salary predictions for future years.