Question
The balance in a savings account at the end of each year is shown below. The year is denoted by n, and the account balance is denoted by
n A(n)
1 $110.00
2 $121.00
3 $133.10
4 $146.41
If the bank pays 10% compound interest, what function represents the total amount in the account in year n?
n A(n)
1 $110.00
2 $121.00
3 $133.10
4 $146.41
If the bank pays 10% compound interest, what function represents the total amount in the account in year n?
Answers
GPT-4o mini
The balance in a savings account with compound interest can be represented by the formula:
\[
A(n) = P(1 + r)^n
\]
where:
- \( A(n) \) is the amount in the account at the end of year \( n \),
- \( P \) is the principal amount (the initial balance),
- \( r \) is the annual interest rate (as a decimal),
- \( n \) is the number of years.
In this case, we can see from the balances provided that the interest rate \( r \) is 10%, or \( 0.1 \).
To find the principal \( P \), we can look at the balance after the first year, which is \( A(1) = 110.00 \). Thus we can set up the equation:
\[
110 = P(1 + 0.1)^1
\]
This simplifies to:
\[
110 = P(1.1)
\]
Solving for \( P \):
\[
P = \frac{110}{1.1} = 100
\]
Now that we have \( P = 100 \), we can express the account balance after \( n \) years using the compound interest formula:
\[
A(n) = 100(1.1)^n
\]
Therefore, the function that represents the total amount in the account in year \( n \) is:
\[
A(n) = 100(1.1)^n
\]
\[
A(n) = P(1 + r)^n
\]
where:
- \( A(n) \) is the amount in the account at the end of year \( n \),
- \( P \) is the principal amount (the initial balance),
- \( r \) is the annual interest rate (as a decimal),
- \( n \) is the number of years.
In this case, we can see from the balances provided that the interest rate \( r \) is 10%, or \( 0.1 \).
To find the principal \( P \), we can look at the balance after the first year, which is \( A(1) = 110.00 \). Thus we can set up the equation:
\[
110 = P(1 + 0.1)^1
\]
This simplifies to:
\[
110 = P(1.1)
\]
Solving for \( P \):
\[
P = \frac{110}{1.1} = 100
\]
Now that we have \( P = 100 \), we can express the account balance after \( n \) years using the compound interest formula:
\[
A(n) = 100(1.1)^n
\]
Therefore, the function that represents the total amount in the account in year \( n \) is:
\[
A(n) = 100(1.1)^n
\]