Question 1

Question

Create 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 Sequence

• Sequence: 2, 5, 8, 11, 14, 17

• Explanation: This sequence is arithmetic because each term after the first is derived by adding a constant difference, which is 3 in this case.

• Fifth Term: a₅ = 14

• Explicit Formula: The explicit formula for the arithmetic sequence can be written as: \[ a_n = a_1 + (n-1)d \] where \( a_1 = 2 \) (first term) and \( d = 3 \) (common difference). Thus, \[ a_n = 2 + (n-1) \cdot 3 = 3n - 1 \]

• Finding the 100th Term: \[ a_{100} = 3(100) - 1 = 300 - 1 = 299 \]

Geometric Sequence

• Sequence: 3, 6, 12, 24, 48, 96

• Explanation: This sequence is geometric because each term after the first is derived by multiplying the previous term by a constant ratio, which is 2 in this case.

• Recursive Formula: \[ a_n = 2 \cdot a_{n-1} \text{ for } n \geq 2 \] with \( a_1 = 3 \).

• Explicit Formula: \[ a_n = a_1 \cdot r^{(n-1)} \] where \( a_1 = 3 \) and \( r = 2 \). Thus, \[ a_n = 3 \cdot 2^{(n-1)} \]

• Finding the 100th Term: \[ a_{100} = 3 \cdot 2^{99} \]

Question 2

• Graphing the Arithmetic Sequence: You would plot the points (1,2), (2,5), (3,8), (4,11), (5,14), (6,17) on graph paper and connect them to form a straight line.

• Linear Function: The linear function representing the arithmetic sequence is: \[ f(x) = 3x - 1 \] This same structure as \( a_n = 3n - 1 \).

• Graphing the Geometric Sequence: You would plot the points (1,3), (2,6), (3,12), (4,24), (5,48), (6,96) on graph paper.

• Exponential Function: The exponential function representing the geometric sequence is: \[ f(x) = 3 \cdot 2^{(x-1)} \] This matches the structure of \( a_n = 3 \cdot 2^{(n-1)} \).

Question 3

• Chosen Amount: Carly receives $10 every week.

• Explanation: Carly's savings grow linearly because she adds the same fixed amount every week, resulting in a straight line when graphed.

• Sequence of Savings: \( 10, 20, 30, 40, 50, \ldots \) (adding 10 each week for 20 weeks).

• Formula for Savings: \[ S_n = 10n \]

• Graph: The points plotted would be (1,10), (2,20), (3,30), …, (20,200). The graph is a straight line indicating a linear relationship.

Question 4

• Chosen Amount: Nico's grandparents put in $1,000.

• Explanation: The amount in the account grows exponentially because it increases by a constant percentage, 8%, each year.

• Sequence of Savings: \( 1000, 1080, 1166.40, 1259.71, \ldots \)

• Formula: \[ A_n = 1000 \cdot (1.08)^{n-1} \]

• Graph: The points plotted will display a curve that rises quickly as time progresses, indicating exponential growth.

Question 5

• Chosen Amount: Aria buys a car for $20,000.

• Explanation: The value of Aria's car decreases exponentially because it loses a constant percentage, 15%, of its value each year.

• Sequence of Car Value: \( 20000, 17000, 14450, 12282.50, \ldots \)

• Formula: \[ V_n = 20000 \cdot (0.85)^{n-1} \]

• Graph: The points plotted would show a downward curve, depicting exponential decay.

Question 6

• Linear Functions: In linear functions, the rate of growth is constant; for example, in Carly's savings, she saves $10 every week, leading to a straight line graph.

• Exponential Functions: In contrast, exponential functions grow by increasing percentages, which means they accelerate over time; for instance, Nico's savings increase by 8% each year, leading to a curve that steepens over time.

• Statement Explanation: The statement indicates that while linear growth increases by a constant amount, exponential growth increases by a percentage of the current amount, leading to much larger values as time goes on.

In comparing Carly's linear savings to Nico's exponential growth, you can see that initially, the amounts may be similar, but over time, Nico's amount grows significantly larger due to the compounded interest, illustrating that the growth rate of an exponential function will eventually exceed any linear growth.