Suppose you deposit $2343 into an account at the end of each year for the next 10 years and then you stop making deposits. If the account earns 6.7% (assume annual compounding), how much will be in the account at the end of 30 years?

To calculate the amount in the account at the end of 30 years, we need to break down the problem into two parts:

1. **Calculate the amount in the account after the first 10 years**, during which deposits are being made.
2. **Calculate the future value of that amount** over the remaining 20 years, during which no further deposits are made.

### Step 1: Calculate the amount after the first 10 years
This involves calculating the future value of an annuity. The formula for the future value of an annuity (FV) is:

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

Where:
- $P$ = Annual payment (deposit) = $2343
- $r$ = Annual interest rate = 6.7% = 0.067
- $n$ = Number of periods (years) = 10

Substituting the values:

\[
FV_{10} = 2343 \times \frac{(1 + 0.067)^{10} - 1}{0.067}
\]

### Step 2: Calculate the amount after the next 20 years
The amount accumulated after 10 years will grow for another 20 years with no additional deposits. This is simply the future value of a single sum, given by the formula:

\[
FV_{30} = FV_{10} \times (1 + r)^{20}
\]

Now, let's calculate these values.

### Calculation

1. **Future Value after 10 years (FV$_{10}$)**:
   \[
   FV_{10} = 2343 \times \frac{(1 + 0.067)^{10} - 1}{0.067} 
   \]

2. **Future Value after 30 years (FV$_{30}$)**:
   \[
   FV_{30} = FV_{10} \times (1 + 0.067)^{20}
   \]

Let's calculate the exact values.1. **Future Value after 10 years (FV$_{10}$)**: \$31,916.84

2. **Future Value after 30 years (FV$_{30}$)**: \$116,763.83

So, at the end of 30 years, the amount in the account will be **\$116,763.83**.

Would you like further details or have any questions?

### Additional Questions:
1. How would the future value change if the interest rate were 5% instead of 6.7%?
2. What if the deposits were made at the beginning of each year instead of the end?
3. How does the future value change if the number of years for deposits increases to 15 years?
4. What would be the total amount of deposits made over the 10 years?
5. How would the outcome change if the interest were compounded monthly instead of annually?
6. What is the effect of increasing the annual deposit amount by 10% each year?
7. How does the present value of the account compare to the future value?
8. How much interest is earned over the entire 30-year period?

### Tip:
Always consider the impact of compounding frequency on the future value of an investment. The more frequent the compounding, the higher the future value.

Learn how to calculate the future value of an annuity with annual deposits and compound interest. This problem involves determining the total amount in an account after 30 years with $2343 deposited annually at 6.7% interest. Suitable for college-level finance and mathematics studies.

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