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.
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