Question
Willow is in her fourth year working for the same company. Her starting salary was $48,000. The next year, her salary was $50,400. The following year, her salary was $52,920. This year, her salary is $55,566. Willow’s boss says her salary will continue to increase by the same percentage each year.
Write the values given in the scenario as a sequence. 48,000, 50,400, 52,920, 55,566,
create a function that models this sequence/ scenario using the geometric explicit formula
An=a x r^( n - 1)
By what percentage is Willow's salary increasing each year ?show your work?
Willow wants to use your function to predict her salary in the future she wants to make sure her predictions are made to reasonable level of accuracy how can she do this?
Write the values given in the scenario as a sequence. 48,000, 50,400, 52,920, 55,566,
create a function that models this sequence/ scenario using the geometric explicit formula
An=a x r^( n - 1)
By what percentage is Willow's salary increasing each year ?show your work?
Willow wants to use your function to predict her salary in the future she wants to make sure her predictions are made to reasonable level of accuracy how can she do this?
Answers
GPT-4o mini
To analyze Willow's salary progression, let's begin by determining the sequence of salaries:
1. Starting Salary (Year 1): \( A_1 = 48,000 \)
2. Year 2 Salary: \( A_2 = 50,400 \)
3. Year 3 Salary: \( A_3 = 52,920 \)
4. Year 4 Salary: \( A_4 = 55,566 \)
### Step 1: Calculate the Common Ratio (r)
The salary appears to grow by a percentage each year, indicating a geometric sequence. To find the rate of increase (the common ratio, \( r \)), we can calculate the ratio of each year's salary to the previous year's salary:
1. From Year 1 to Year 2:
\[
r = \frac{A_2}{A_1} = \frac{50,400}{48,000} = 1.05
\]
2. From Year 2 to Year 3:
\[
r = \frac{A_3}{A_2} = \frac{52,920}{50,400} = 1.05
\]
3. From Year 3 to Year 4:
\[
r = \frac{A_4}{A_3} = \frac{55,566}{52,920} \approx 1.05
\]
The common ratio \( r \) is consistently approximately \( 1.05 \).
### Step 2: Identify the Rate of Increase
To find the percentage increase in Willow's salary each year:
\[
\text{Percentage Increase} = (r - 1) \times 100\%
\]
\[
\text{Percentage Increase} = (1.05 - 1) \times 100\% = 0.05 \times 100\% = 5\%
\]
Willow's salary increases by **5%** each year.
### Step 3: Write the Geometric Explicit Formula
Given the explicit formula for a geometric sequence:
\[
A_n = A_1 \cdot r^{(n - 1)}
\]
Substituting \( A_1 = 48,000 \) and \( r = 1.05 \):
\[
A_n = 48,000 \cdot (1.05)^{(n - 1)}
\]
### Step 4: Prediction of Future Salaries
To predict Willow's salary in the future, she can utilize the explicit function \( A_n = 48,000 \cdot (1.05)^{(n - 1)} \) for any year \( n \).
### Step 5: Ensuring Predictions are Accurate
1. **Review the Assumption**: Willow should monitor her salary pattern closely. If any changes occur in company policy, performance evaluations, or external economic factors, it may impact future salary growth.
2. **Use Predictions with a Margin of Error**: Variables like promotions, job changes, or economic conditions may alter salary growth. For more accuracy, she might consider conservative projections and include a range based on potential salary increases (lower and upper bounds).
3. **Regularly Update**: As time goes on, she should use actual salary updates to adjust her predictions and ensure her model remains accurate.
4. **Consult HR**: If possible, Willow should talk to HR about potential raises, bonuses, or other elements affecting salary progression.
By following these guidelines, Willow can utilize her function effectively to forecast reasonable salary predictions for her future.
1. Starting Salary (Year 1): \( A_1 = 48,000 \)
2. Year 2 Salary: \( A_2 = 50,400 \)
3. Year 3 Salary: \( A_3 = 52,920 \)
4. Year 4 Salary: \( A_4 = 55,566 \)
### Step 1: Calculate the Common Ratio (r)
The salary appears to grow by a percentage each year, indicating a geometric sequence. To find the rate of increase (the common ratio, \( r \)), we can calculate the ratio of each year's salary to the previous year's salary:
1. From Year 1 to Year 2:
\[
r = \frac{A_2}{A_1} = \frac{50,400}{48,000} = 1.05
\]
2. From Year 2 to Year 3:
\[
r = \frac{A_3}{A_2} = \frac{52,920}{50,400} = 1.05
\]
3. From Year 3 to Year 4:
\[
r = \frac{A_4}{A_3} = \frac{55,566}{52,920} \approx 1.05
\]
The common ratio \( r \) is consistently approximately \( 1.05 \).
### Step 2: Identify the Rate of Increase
To find the percentage increase in Willow's salary each year:
\[
\text{Percentage Increase} = (r - 1) \times 100\%
\]
\[
\text{Percentage Increase} = (1.05 - 1) \times 100\% = 0.05 \times 100\% = 5\%
\]
Willow's salary increases by **5%** each year.
### Step 3: Write the Geometric Explicit Formula
Given the explicit formula for a geometric sequence:
\[
A_n = A_1 \cdot r^{(n - 1)}
\]
Substituting \( A_1 = 48,000 \) and \( r = 1.05 \):
\[
A_n = 48,000 \cdot (1.05)^{(n - 1)}
\]
### Step 4: Prediction of Future Salaries
To predict Willow's salary in the future, she can utilize the explicit function \( A_n = 48,000 \cdot (1.05)^{(n - 1)} \) for any year \( n \).
### Step 5: Ensuring Predictions are Accurate
1. **Review the Assumption**: Willow should monitor her salary pattern closely. If any changes occur in company policy, performance evaluations, or external economic factors, it may impact future salary growth.
2. **Use Predictions with a Margin of Error**: Variables like promotions, job changes, or economic conditions may alter salary growth. For more accuracy, she might consider conservative projections and include a range based on potential salary increases (lower and upper bounds).
3. **Regularly Update**: As time goes on, she should use actual salary updates to adjust her predictions and ensure her model remains accurate.
4. **Consult HR**: If possible, Willow should talk to HR about potential raises, bonuses, or other elements affecting salary progression.
By following these guidelines, Willow can utilize her function effectively to forecast reasonable salary predictions for her future.