at the end of 8 years, Beth will have approximately $138,651.60.

UpStudy Free Solution:

We need to find the future value of an ordinary annuity where Beth deposits $1,000 at the end of each month into an account that pays 9% annual interest, compounded monthly, over 8 years.

The formula for the future value of an ordinary annuity is:

\[FV = P \frac { ( 1 + r) ^ n - 1} { r} \]

where:

- \(FV\) is the future value,

- \(P\) is the payment amount per period ($1,000),

- \(r\) is the monthly interest rate,

- \(n\) is the total number of payments.

Given:

- \(P = 1000\),

- Annual interest rate = 9%, so the monthly interest rate \(r = \frac { 9\% } { 12} = 0.75\% = 0.0075\),

- Number of years = 8, so the total number of monthly payments \(n = 8 \times 12 = 96\).

Now, plug these values into the formula:

\[FV = 1000 \frac { ( 1 + 0.0075) ^ { 96} - 1} { 0.0075} \]

First, calculate \(( 1 + 0.0075) ^ { 96} \):

\[( 1 + 0.0075) ^ { 96} \approx 2.039887\]

Next, calculate the numerator:

\[2.039887 - 1 = 1.039887\]

Now, divide by the monthly interest rate:

\[\frac { 1.039887} { 0.0075} \approx 138.6516\]

Finally, multiply by the payment amount:

\[FV = 1000 \times 138.6516 \approx 138,651.60\]

Supplemental Knowledge

In financial mathematics, understanding how to calculate the future value of an annuity is crucial for planning investments and savings. Here’s a deeper look into the concepts involved:

1. Future Value of an Ordinary Annuity:

- An ordinary annuity involves making equal payments at the end of each period. The future value (FV) represents the total amount accumulated after all payments have been made, including interest.

- The formula for the future value of an ordinary annuity is:

\[FV = P \frac { ( 1 + r) ^ n - 1} { r} \]

where:

- \(FV\) is the future value,

- \(P\) is the payment amount per period,

- \(r\) is the interest rate per period,

- \(n\) is the total number of payments.

2. Interest Rate Per Period:

- When dealing with monthly payments, convert the annual interest rate to a monthly rate by dividing by 12.

- For example, an annual interest rate of 9% becomes a monthly rate of \(0.75\% \) or \(0.0075\).

3. Number of Payments:

- The total number of payments (\(n\)) is calculated by multiplying the number of years by the number of periods per year.

- For example, for 8 years with monthly payments, \(n = 8 \times 12 = 96\).

4. Compound Interest:

- Compound interest plays a significant role in growing investments over time. Each payment earns interest on both principal and previously earned interest.

Planning your financial future can be complex. Mastering these calculations ensures you make informed decisions regarding investments and savings accounts. UpStudy offers additional help or can deepen your understanding of financial mathematics for greater financial confidence.

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