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.

Question

explain why the money in nicos saving account over time is an example of exponential growth

Nico's saving account is an example of exponential growth because it is ___ each year by a factor of ____

create a sequence that describes the amount of money in nicos savings account each year by 8 years (write answers in this format 2,000 or 1,456.78

create a formula for the amount of money in nicos savings account over time
y=a*b^x
a= initial amount
b= growth factor

y=__(__)^x

use your formula to find the amount of money in nicos savings account after 18 years.
after 18 years nico has $___ in his savings account

Answer 1

Nico's savings account is an example of exponential growth because it is increased each year by a factor of 1.08 (which represents the 8% growth).

Sequence of Amounts in Nico's Savings Account Over Time

To calculate the amount of money in the savings account each year up to 8 years, we can use the formula for exponential growth:

1. Year 0 (initial amount): \(1000\)
2. Year 1: \(1000 \times 1.08 = 1080\)
3. Year 2: \(1000 \times (1.08)^2 \approx 1166.40\)
4. Year 3: \(1000 \times (1.08)^3 \approx 1259.71\)
5. Year 4: \(1000 \times (1.08)^4 \approx 1360.49\)
6. Year 5: \(1000 \times (1.08)^5 \approx 1469.32\)
7. Year 6: \(1000 \times (1.08)^6 \approx 1586.87\)
8. Year 7: \(1000 \times (1.08)^7 \approx 1713.74\)
9. Year 8: \(1000 \times (1.08)^8 \approx 1850.93\)

So the sequence is: 1,000, 1,080, 1,166.40, 1,259.71, 1,360.49, 1,469.32, 1,586.87, 1,713.74, 1,850.93

Formula for the Amount of Money in Nico's Savings Account

The general formula for exponential growth can be expressed as:

\[ y = a \cdot b^x \]

Where:

• \( a = 1000 \) (initial amount)
• \( b = 1.08 \) (growth factor)

Thus, the formula for Nico's savings account is:

\[ y = 1000 \cdot (1.08)^x \]

Amount After 18 Years

To find the amount in Nico's savings account after 18 years, we substitute \( x = 18 \) into our formula:

\[ y = 1000 \cdot (1.08)^{18} \]

Calculating \( (1.08)^{18} \):

\[ (1.08)^{18} \approx 3.9968 \]

Now substituting back into the formula:

\[ y \approx 1000 \cdot 3.9968 \approx 3996.80 \]

After 18 years, Nico has approximately $3,996.80 in his savings account.