You just deposited $5,500 in an account.  You plan on depositing $600 at the end of every month for the next 13 years.  How much will you have if the account pays a monthly interest rate of 0.5%?

To determine how much money you will have after 13 years with an initial deposit of $5,500 and monthly deposits of $600 at a monthly interest rate of 0.5%, we need to calculate the future value of both the initial deposit and the series of monthly deposits. 

### Part 1: Future Value of the Initial Deposit

The future value (FV) of a single sum of money is calculated using the formula:
\[ FV_{\text{initial}} = PV \times (1 + r)^n \]
where:
- $ PV $ is the present value ($5,500),
- $ r $ is the monthly interest rate (0.5% or 0.005),
- $ n $ is the number of months (13 years $\times$ 12 months/year = 156 months).

Substitute the values:
\[ FV_{\text{initial}} = 5500 \times (1 + 0.005)^{156} \]

### Part 2: Future Value of Monthly Deposits

The future value of an annuity (series of regular deposits) is calculated using the formula:
\[ FV_{\text{annuity}} = P \times \frac{(1 + r)^n - 1}{r} \]
where:
- $ P $ is the monthly deposit amount ($600),
- $ r $ is the monthly interest rate (0.5% or 0.005),
- $ n $ is the number of months (156 months).

Substitute the values:
\[ FV_{\text{annuity}} = 600 \times \frac{(1 + 0.005)^{156} - 1}{0.005} \]

### Calculation

Let's calculate each part step-by-step.

#### Part 1: Future Value of the Initial Deposit
1. Calculate $ (1 + 0.005)^{156} $:
\[ (1 + 0.005)^{156} \approx 2.262 \]

2. Multiply by the initial deposit:
\[ FV_{\text{initial}} = 5500 \times 2.262 \approx 12,441 \]

#### Part 2: Future Value of Monthly Deposits
1. Calculate $ (1 + 0.005)^{156} - 1 $:
\[ (1 + 0.005)^{156} - 1 \approx 1.262 \]

2. Divide by the interest rate (0.005):
\[ \frac{1.262}{0.005} \approx 252.4 \]

3. Multiply by the monthly deposit amount:
\[ FV_{\text{annuity}} = 600 \times 252.4 \approx 151,440 \]

### Total Future Value
Add the future value of the initial deposit and the future value of the annuity:
\[ FV_{\text{total}} = FV_{\text{initial}} + FV_{\text{annuity}} \approx 12,441 + 151,440 \approx 163,881 \]

Therefore, after 13 years, you will have approximately $163,881 in the account.

Do you want more details or have any questions?

Here are some related questions you might find useful:
1. How do different compounding periods affect the future value of an investment?
2. What is the difference between future value and present value?
3. How can you calculate the future value of an uneven series of deposits?
4. What impact do higher monthly deposits have on the future value of an account?
5. How does varying the interest rate affect the future value of an investment?

**Tip:** Consistently making deposits and understanding the power of compound interest can significantly increase your savings over time.

Learn how to calculate the future value of an investment with an initial deposit of $5,500 and monthly deposits of $600 over 13 years, with a monthly interest rate of 0.5%. Understand the formulas for future value of a single sum and future value of an annuity. Approximate total value: $163,881.

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