Certainly! Let’s go through each part of your request step by step.
1. Choose a dollar amount
Let’s say Nico’s grandparents put $1,000 in the savings account when he was born.
2. Explanation of exponential growth
The money in Nico’s savings account grows exponentially over time because of the interest that is earned on the initial principal and subsequently on the accumulated interest. If the account earns compound interest, the total amount in the account increases at a rate proportional to its current value. For example, if the account earns a fixed annual interest rate, say 5%, the interest earned during the first year would be 5% of $1,000, and in the second year, it would be 5% of the new total, which includes the interest from the previous year (i.e., $1,000 + interest).
3. Sequence describing the money in Nico's savings account
Assuming a 5% annual interest rate, we can create a sequence for each year:
- Year 0: \( A_0 = 1000 \)
- Year 1: \( A_1 = A_0 \times (1 + r) = 1000 \times (1 + 0.05) = 1000 \times 1.05 = 1050 \)
- Year 2: \( A_2 = A_1 \times (1 + r) = 1050 \times 1.05 = 1102.50 \)
- Year 3: \( A_3 = A_2 \times (1 + r) = 1102.50 \times 1.05 = 1157.63 \)
- Continuing this for 20 years...
Here’s the full sequence for 20 years, rounded for simplicity:
- Year 1: $1,050.00
- 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: $1,551.33
- Year 10: $1,628.90
- Year 11: $1,710.35
- Year 12: $1,795.86
- Year 13: $1,885.65
- Year 14: $1,980.93
- Year 15: $2,081.98
- Year 16: $2,189.08
- Year 17: $2,302.54
- Year 18: $2,422.66
- Year 19: $2,549.79
- Year 20: $2,684.28
4. Formula for the savings account over time
The formula for the amount of money in Nico’s savings account can be expressed using the compound interest formula:
\[ A = P \times (1 + r)^t \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money, which is $1,000 in this case).
- \( r \) is the annual interest rate (decimal) (5% or 0.05).
- \( t \) is the time the money is invested for in years.
For this example: \[ A = 1000 \times (1 + 0.05)^t \]
5. Graphing and key features
To create a graph for this scenario, you would plot the years on the x-axis (from 0 to 20) and the amount of money on the y-axis (from $1,000 to about $2,700).
Key features of the graph that signal it depicts exponential growth:
- Curvature: The graph of the amount of money increases slowly at first and then steepens, demonstrating that as time progresses, the growth rate increases. This characteristic, where the growth accelerates, is a hallmark of exponential functions.
- Steepness Increase: At the beginning of the graph, the increases are relatively small, but as time goes on, the increases become much larger, creating a steep upward curve.
Note:
Remember, when plotting on graph paper or digitally, the growth pattern shows the characteristic 'J-shaped' curve of exponential growth.
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