Question

1: Choose a dollar amount that you think Nico’s grandparents might have put in
the savings account when Nico was born. (Answered, $1,000.)

2: Explain why the money in Nico’s savings account over time is an example of
exponential growth.

3: Create a sequence that describes the amount of money in Nico’s savings
account each year for 20 years.

4: Create a formula for the amount of money in Nico’s savings account over time.

5: 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.

Make the answers as short as possible.
Answer each question seperately

Answer 1

1: **Dollar Amount Choice**
$1,000.

2: **Explanation of Exponential Growth**
The money in Nico's savings account grows exponentially because it earns interest on both the initial principal and the accumulated interest, leading to a growth rate that increases over time.

3: **Amount of Money Each Year for 20 Years**
If the account earns an annual interest rate of 5%, the sequence for the amount of money in the account each year is:
Year 0: $1,000
Year 1: $1,050
Year 2: $1,102.50
Year 3: $1,157.63
Year 4: $1,215.51
Year 5: $1,276.28
Year 6: $1,340.10
Year 7: $1,407.10
Year 8: $1,477.45
Year 9: $1,551.32
Year 10: $1,628.89
Year 11: $1,710.34
Year 12: $1,795.86
Year 13: $1,885.64
Year 14: $1,979.92
Year 15: $2,078.92
Year 16: $2,182.87
Year 17: $2,292.01
Year 18: $2,406.61
Year 19: $2,526.94
Year 20: $2,653.27

4: **Formula for Amount Over Time**
The formula is \( A = P(1 + r)^t \), where:
- \( A \) = amount in the account after \( t \) years
- \( P \) = initial principal ($1,000)
- \( r \) = annual interest rate (0.05)
- \( t \) = number of years

Thus, \( A = 1000(1 + 0.05)^t \).

5: **Graph and Key Feature**
To create the graph, plot the values from Year 0 to Year 20 using the formula. A key feature is the curve of the graph, which steepens over time, indicating that the rate of growth accelerates, a hallmark of exponential growth.