When Nico was born, his grandparents put $3,500 in a special savings account. The amount of money in the savings account increases by 4% APY (annual percentage yield). Nico tracks how much money is in the account over time.

Answer 1

If Nico's grandparents deposited $3,500 in a savings account that earns a 4% annual percentage yield (APY), we can calculate the amount of money in the account after a certain number of years using the formula for compound interest:

\[ A = P(1 + r)^t \]

Where:

• \( A \) is the amount of money accumulated after n years, including interest.
• \( P \) is the principal amount (the initial amount of money).
• \( r \) is the annual interest rate (decimal).
• \( t \) is the number of years the money is invested or borrowed.

Given:

• \( P = 3500 \)
• \( r = 0.04 \)

Let's calculate the amount in the account after different time periods. Here are a few examples:

1. After 1 Year: \[ A = 3500(1 + 0.04)^1 = 3500(1.04) = 3640 \]

2. After 5 Years: \[ A = 3500(1 + 0.04)^5 = 3500(1.04)^5 \approx 3500(1.2167) \approx 4264.45 \]

3. After 10 Years: \[ A = 3500(1 + 0.04)^{10} = 3500(1.04)^{10} \approx 3500(1.4802) \approx 5170.70 \]

4. After 20 Years: \[ A = 3500(1 + 0.04)^{20} = 3500(1.04)^{20} \approx 3500(2.2080) \approx 7728.00 \]

If you want to calculate the amount for a different number of years, just plug in the value of \( t \) into the formula.

Answer 2

Explain why the money in Nico’s savings account over time is an example of exponential growth. (1 pts)

Answer 3

The money in Nico's savings account is an example of exponential growth because the amount of money increases by a fixed percentage (4% in this case) each year, which means the growth is relative to the current amount in the account.

In exponential growth, the value of a quantity increases at a rate proportional to its current value, resulting in the growth rate itself increasing over time.

Here's why this is the case in Nico's situation:

1. Percentage Increase: Each year, Nico's savings account earns interest based on the total amount already in the account, not just the initial deposit. For example, after the first year, the interest is calculated on $3,500, but in the second year, the interest is calculated on $3,640 (the initial amount plus the interest earned in the first year). This compounding effect leads to larger increments of growth each year.

2. Mathematical Representation: The formula used to calculate the account balance, \( A = P(1 + r)^t \), features an exponent \( t \) which reflects this compounding. This indicates that as time increases, the amount \( A \) will grow significantly more than it would with linear growth—where the increase would just be a constant dollar amount.

3. Graphing the Growth: If you were to graph the growth of the savings account over time, you would see a curve that rises steeply, illustrating how the amount accelerates as time goes on, characteristic of exponential growth.

In summary, because the interest earned builds on the previously earned interest, the account grows faster and faster over time, fulfilling the definition of exponential growth.

Answer 4

simple

Answer 5

Nico's savings account shows exponential growth because the money increases by a fixed percentage (4%) each year. This means that each year, the interest earned is calculated on the total amount in the account, which gets larger over time. As a result, the growth rate itself increases, making the amount of money grow faster and faster. This is why the growth is exponential rather than linear.

Answer 6

Create a sequence that describes the amount of money in Nico’s savings account each year for 20 years. (Show all your work). (2 pts)

Answer 7

To create a sequence that describes the amount of money in Nico's savings account each year for 20 years using the compound interest formula, we will calculate the value for each year from 1 to 20.

The formula we will use is: \[ A = P(1 + r)^t \] Where:

• \( A \) = amount after \( t \) years
• \( P = 3500 \) (the initial amount)
• \( r = 0.04 \) (the annual interest rate)
• \( t \) = number of years

Let’s calculate the balance for each year:

1. Year 1: \[ A_1 = 3500(1 + 0.04)^1 = 3500(1.04) = 3640 \]

2. Year 2: \[ A_2 = 3500(1 + 0.04)^2 = 3500(1.04)^2 = 3500(1.0816) \approx 3785.60 \]

3. Year 3: \[ A_3 = 3500(1 + 0.04)^3 = 3500(1.04)^3 = 3500(1.124864) \approx 3937.02 \]

4. Year 4: \[ A_4 = 3500(1 + 0.04)^4 = 3500(1.04)^4 = 3500(1.16985856) \approx 4094.50 \]

5. Year 5: \[ A_5 = 3500(1 + 0.04)^5 = 3500(1.04)^5 = 3500(1.2166529024) \approx 4263.29 \]

6. Year 6: \[ A_6 = 3500(1 + 0.04)^6 = 3500(1.04)^6 = 3500(1.265319) \approx 4428.63 \]

7. Year 7: \[ A_7 = 3500(1 + 0.04)^7 = 3500(1.04)^7 = 3500(1.315928) \approx 4596.75 \]

8. Year 8: \[ A_8 = 3500(1 + 0.04)^8 = 3500(1.04)^8 = 3500(1.368569) \approx 4769.00 \]

9. Year 9: \[ A_9 = 3500(1 + 0.04)^9 = 3500(1.04)^9 = 3500(1.423316) \approx 4971.60 \]

10. Year 10: \[ A_{10} = 3500(1 + 0.04)^{10} = 3500(1.04)^{10} = 3500(1.48024) \approx 5188.84 \]

11. Year 11: \[ A_{11} = 3500(1 + 0.04)^{11} = 3500(1.04)^{11} \approx 3500(1.53984) \approx 5389.44 \]

12. Year 12: \[ A_{12} = 3500(1 + 0.04)^{12} = 3500(1.04)^{12} \approx 3500(1.60103) \approx 5603.16 \]

13. Year 13: \[ A_{13} = 3500(1 + 0.04)^{13} = 3500(1.04)^{13} \approx 3500(1.66409) \approx 5822.32 \]

14. Year 14: \[ A_{14} = 3500(1 + 0.04)^{14} = 3500(1.04)^{14} \approx 3500(1.72876) \approx 6053.66 \]

15. Year 15: \[ A_{15} = 3500(1 + 0.04)^{15} = 3500(1.04)^{15} \approx 3500(1.79523) \approx 6289.36 \]

16. Year 16: \[ A_{16} = 3500(1 + 0.04)^{16} = 3500(1.04)^{16} \approx 3500(1.86342) \approx 6519.25 \]

17. Year 17: \[ A_{17} = 3500(1 + 0.04)^{17} = 3500(1.04)^{17} \approx 3500(1.93343) \approx 6771.997 \]

18. Year 18: \[ A_{18} = 3500(1 + 0.04)^{18} = 3500(1.04)^{18} \approx 3500(2.00525) \approx 7017.54 \]

19. Year 19: \[ A_{19} = 3500(1 + 0.04)^{19} = 3500(1.04)^{19} \approx 3500(2.07809) \approx 7267.33 \]

20. Year 20: \[ A_{20} = 3500(1 + 0.04)^{20} = 3500(1.04)^{20} \approx 3500(2.152017) \approx 7487.37 \]

Summary of Amounts for Each Year:

• Year 1: $3,640.00
• Year 2: $3,785.60
• Year 3: $3,937.02
• Year 4: $4,094.50
• Year 5: $4,263.29
• Year 6: $4,428.63
• Year 7: $4,596.75
• Year 8: $4,769.00
• Year 9: $4,971.60
• Year 10: $5,188.84
• Year 11: $5,389.44
• Year 12: $5,603.16
• Year 13: $5,822.32
• Year 14: $6,053.66
• Year 15: $6,289.36
• Year 16: $6,519.25
• Year 17: $6,771.99
• Year 18: $7,017.54
• Year 19: $7,267.33
• Year 20: $7,487.37

This sequence shows the amount in the savings account at the end of each of the 20 years.

Answer 8

just the numbers

Answer 9

Sure! Here are the amounts in Nico's savings account at the end of each year for 20 years:

1. Year 1: $3,640.00
2. Year 2: $3,785.60
3. Year 3: $3,937.02
4. Year 4: $4,094.50
5. Year 5: $4,263.29
6. Year 6: $4,428.63
7. Year 7: $4,596.75
8. Year 8: $4,769.00
9. Year 9: $4,971.60
10. Year 10: $5,188.84
11. Year 11: $5,389.44
12. Year 12: $5,603.16
13. Year 13: $5,822.32
14. Year 14: $6,053.66
15. Year 15: $6,289.36
16. Year 16: $6,519.25
17. Year 17: $6,771.99
18. Year 18: $7,017.54
19. Year 19: $7,267.33
20. Year 20: $7,487.37

Answer 10

Create a formula for the amount of money in Nico’s savings account over time. (Show all your work) simple

Answer 11

To create a formula for the amount of money in Nico's savings account over time, we use the formula for compound interest. The general formula for compound interest is:

\[ A = P(1 + r)^t \]

Where:

• \( A \) is the amount of money accumulated after \( t \) years, including interest.
• \( P \) is the principal amount (the initial amount of money).
• \( r \) is the annual interest rate (in decimal).
• \( t \) is the number of years the money is invested.

Given Values

• \( P = 3500 \) (the initial deposit)
• \( r = 0.04 \) (the annual interest rate of 4%)

Substituting Values into the Formula

Substituting \( P \) and \( r \) into the formula gives:

\[ A = 3500(1 + 0.04)^t \]

This simplifies to:

\[ A = 3500(1.04)^t \]

Final Formula

So, the formula for the amount of money in Nico's savings account after \( t \) years is:

\[ A = 3500(1.04)^t \]

Explanation

• This formula allows you to calculate how much money will be in the account after any number of years (\( t \)), by plugging in the desired value for \( t \).
• The term \( (1.04)^t \) represents the growth factor due to the annual interest being compounded each year.