When Nico was born his grandparents put $1000 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.

Create a sequence that describes the amount of money in Nico’s savings each year for 8 years. (Write answers in the form or 2,000 or 1,567.87)

1 answer

To describe the amount of money in Nico's savings account over time, we can use 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 for.

In this case:

  • \( P = 1000 \)
  • \( r = 0.08 \) (which is 8%)
  • \( t \) will vary from 0 to 8.

We'll calculate the amount for each year from 0 to 8:

  1. Year 0 (initial deposit): \[ A = 1000 \times (1 + 0.08)^0 = 1000 \times 1 = 1000 \]

  2. Year 1: \[ A = 1000 \times (1 + 0.08)^1 = 1000 \times 1.08 = 1080 \]

  3. Year 2: \[ A = 1000 \times (1 + 0.08)^2 = 1000 \times 1.1664 = 1166.40 \]

  4. Year 3: \[ A = 1000 \times (1 + 0.08)^3 = 1000 \times 1.259712 = 1259.71 \]

  5. Year 4: \[ A = 1000 \times (1 + 0.08)^4 = 1000 \times 1.36048864 = 1360.49 \]

  6. Year 5: \[ A = 1000 \times (1 + 0.08)^5 = 1000 \times 1.469328 = 1469.33 \]

  7. Year 6: \[ A = 1000 \times (1 + 0.08)^6 = 1000 \times 1.586874 = 1586.87 \]

  8. Year 7: \[ A = 1000 \times (1 + 0.08)^7 = 1000 \times 1.713775 = 1713.78 \]

  9. Year 8: \[ A = 1000 \times (1 + 0.08)^8 = 1000 \times 1.85093 = 1850.93 \]

Now, compiling this into a sequence for each year from 0 to 8:

  • Year 0: 1000.00
  • Year 1: 1080.00
  • Year 2: 1166.40
  • Year 3: 1259.71
  • Year 4: 1360.49
  • Year 5: 1469.33
  • Year 6: 1586.87
  • Year 7: 1713.78
  • Year 8: 1850.93

Thus, the sequence of the amount in Nico’s savings account each year for 8 years is:

1000.00, 1080.00, 1166.40, 1259.71, 1360.49, 1469.33, 1586.87, 1713.78, 1850.93