Question

reate your own arithmetic sequence. List at least six terms of your sequence. Explain what makes it arithmetic. State the fifth term of your arithmetic sequence using subscript notation. Write an explicit formula for your arithmetic sequence. Use your explicit formula to find the 100th term of your sequence. Create your own geometric sequence. List at least six terms of your sequence. Explain what makes it geometric. Write a recursive formula for your geometric sequence. Write an explicit formula for your geometric sequence. Use your explicit formula to find the 100th term of your sequence. Question 2 Graph your arithmetic sequence on a coordinate grid (a sheet of graph paper). Write a linear function to represent your arithmetic sequence. Show that it has the same structure as your explicit formula from Question 1C. Graph your geometric sequence on a coordinate grid (a sheet of graph paper). Write an exponential function to represent your geometric sequence. Show that it has the same structure as your explicit formula from Question 1F. Question 3 Carly receives $ every week for babysitting for her neighbors. She decides to save the money she receives to help pay for a camp she wants to attend next summer. She tracks how much money she saves from babysitting over the course of the year. Choose a dollar amount that you think Carly might receive each week. Explain why Carly’s savings over time is an example of linear change. Create a sequence that describes Carly’s growing savings each week over the course of 20 weeks. Create a formula for Carly’s savings over time. On a sheet of graph paper, create a graph for Carly’s savings over time. Describe at least one key feature on your graph that signals it depicts linear change. Question 4 When Nico was born, his grandparents put $ in a special savings account. The amount of money in the savings account increases by 8% every year. Nico tracks how much money is in the account over time. Choose a dollar amount that you think Nico’s grandparents might have put in the savings account when Nico was born. Explain why the money in Nico’s savings account over time is an example of exponential growth. Create a sequence that describes the amount of money in Nico’s savings account each year for 20 years. Create a formula for the amount of money in Nico’s savings account over time. On a sheet of graph paper, create a graph for the amount of money in Nico’s savings account over time. Describe at least one key feature on your graph that signals it depicts exponential growth. Question 5 Aria buys a new car for $. She learns that every year her car will depreciate in value by 15%. This means that each year her car will be worth 85% of what it was the previous year. Aria tracks how much her car is worth over time. Choose a dollar amount that you think Aria might have purchased her car for. Explain how the amount of money Aria’s car is worth over time is an example of exponential decay. Create a sequence that describes the worth of Aria’s car each year for 10 years. Create a formula for the worth of Aria’s car over time. On a sheet of graph paper, create a graph for the worth of Aria’s car over time. Describe at least one key feature on your graph that signals it depicts exponential decay. Question 6 Explain in your own words how linear functions grow and how exponential functions grow. Explain the statement “No matter what the growth rate of a linear function is, that of an exponential function will always exceed it.” Explain how your linear function from Question 3 and your exponential function from Question 4 demonstrate that this statement is true.

Answer 1

Question 1: Arithmetic and Geometric Sequences

Arithmetic Sequence

1. Sequence: 2, 5, 8, 11, 14, 17
2. Explanation: An arithmetic sequence is defined by having a constant difference between consecutive terms. In this case, the common difference is 3 (5 - 2 = 3; 8 - 5 = 3; etc.).
3. Fifth Term: \( a_5 = 14 \)
4. Explicit Formula: The explicit formula for the \( n \)-th term of an arithmetic sequence can be written as: \[ a_n = a_1 + (n - 1) \cdot d \] where \( a_1 = 2 \) (the first term) and \( d = 3 \) (the common difference). Thus: \[ a_n = 2 + (n - 1) \cdot 3 = 3n - 1 \]
5. 100th Term: To find the 100th term, substitute \( n = 100 \): \[ a_{100} = 3(100) - 1 = 299 \]

Geometric Sequence

1. Sequence: 3, 6, 12, 24, 48, 96
2. Explanation: A geometric sequence is defined by having a constant ratio between consecutive terms. In this case, the common ratio is 2 (6 / 3 = 2; 12 / 6 = 2; etc.).
3. Recursive Formula: \[ a_n = a_{n-1} \cdot r \quad \text{with } a_1 = 3 \text{ and } r = 2 \]
4. Explicit Formula: The explicit formula for the geometric sequence can be written as: \[ a_n = a_1 \cdot r^{n-1} = 3 \cdot 2^{n-1} \]
5. 100th Term: To find the 100th term, substitute \( n = 100 \): \[ a_{100} = 3 \cdot 2^{99} \] This is a very large number, specifically \( 3 \cdot 2^{99} \).

Question 2: Graphing Sequences

1. Graphing Arithmetic Sequence: This sequence would produce a straight line on a coordinate grid, showing the increase at a constant rate. The linear function would be: \[ f(x) = 3x - 1 \] Comparing this with the explicit formula, they both have similar structures involving a variable \( x \).

2. Graphing Geometric Sequence: The geometric sequence would create a curve that rises steeply as \( n \) increases. The exponential function would be: \[ g(x) = 3 \cdot 2^{x-1} \] Again, comparing this with the explicit formula, we see they reflect the same growth nature of an exponential function.

Question 3: Carly's Savings

1. Weekly Earnings: Assume Carly receives $10 every week.
2. Linear Change Explanation: Carly’s total savings after each week increases linearly, as she adds a fixed amount ($10) each week.
3. Sequence: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, ..., 200 (for 20 weeks).
4. Formula: \[ S_n = 10n \text{ (where } n \text{ is the number of weeks)} \]
5. Graph Features: The graph will be a straight line with a positive slope, indicating consistent, linear growth.

Question 4: Nico's Savings Account

1. Initial Deposit: Assume Nico's grandparents put in $1,000.
2. Exponential Growth Explanation: The amount of money in the savings account grows exponentially because it increases by a percentage (8%) annually.
3. Sequence: $1,000, $1,080, $1,166.4 (and so on) for 20 years.
4. Formula for Savings: \[ A_n = 1000 \cdot (1.08)^{n-1} \]
5. Graph Features: The graph will show a curve that accelerates upwards, signaling exponential growth.

Question 5: Aria's Car Depreciation

1. Initial Cost: Assume Aria bought her car for $20,000.
2. Exponential Decay Explanation: The value of the car decreases by a fixed percentage (15%) each year, defining exponential decay.
3. Sequence: $20,000, $17,000, $14,450 (and so on) for 10 years.
4. Formula for Value: \[ V_n = 20000 \cdot (0.85)^{n-1} \]
5. Graph Features: The graph will be a curve that slowly approaches zero, indicating exponential decay.

Question 6: Differences in Growth

• Linear Functions grow by adding a fixed amount, leading to a straight line.
• Exponential Functions grow by a fixed percentage, resulting in curved graphs that get steeper over time.
• The statement about exponential functions always exceeding linear functions is true because, although linear functions have fixed growth, exponential functions compound, causing them to surpass linear growth significantly over time, as seen in Carly's and Nico’s examples where Nico’s growth rapidly exceeds Carly’s after a few years.