You need to accumulate $10,000. To do so, you plan to make deposits of $1,450 per year - with the first payment being made a year from today - into a bank account that pays 6% annual interest. Your last deposit will be less than $1,450 if less is needed to round out to $10,000. How many years will it take you to reach your $10,000 goal? Do not round intermediate calculations. Round your answer up to the nearest whole number.

  year(s)

How large will the last deposit be? Do not round intermediate calculations. Round your answer to the nearest cent.

$   

To solve this problem, we need to determine:

1. How many years it will take to reach the $10,000 goal.
2. The amount of the last deposit if less than $1,450 is needed to reach exactly $10,000.

### Step 1: Calculate the Future Value of an Annuity

The formula for the future value of an annuity is:

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

Where:
- $ P $ is the annual deposit ($1,450)
- $ r $ is the annual interest rate (6% or 0.06)
- $ n $ is the number of years

### Step 2: Estimate the Number of Years

We start by solving for $ n $ in the future value of an annuity formula, knowing that the future value should be $10,000.

\[
10,000 = 1,450 \times \frac{(1 + 0.06)^n - 1}{0.06}
\]

Simplifying the equation:

\[
\frac{10,000}{1,450} = \frac{(1.06)^n - 1}{0.06}
\]

\[
6.89655 = \frac{(1.06)^n - 1}{0.06}
\]

\[
0.413793 = (1.06)^n - 1
\]

\[
(1.06)^n = 1.413793
\]

Now, we need to solve for $ n $ by taking the logarithm of both sides:

\[
n \times \log(1.06) = \log(1.413793)
\]

\[
n = \frac{\log(1.413793)}{\log(1.06)} \approx \frac{0.1504}{0.0253} \approx 5.94
\]

Since we cannot make a partial deposit, we round this up to the next whole number:

\[
n = 6 \text{ years}
\]

### Step 3: Calculate the Exact Amount After 5 Years

Now, let's find out how much you will have after 5 years to determine the amount needed in the 6th year.

\[
\text{Future Value after 5 years} = 1,450 \times \frac{(1.06)^5 - 1}{0.06}
\]

\[
\text{Future Value after 5 years} = 1,450 \times \frac{(1.3382 - 1)}{0.06} \approx 1,450 \times 5.6367 \approx 8,173.21
\]

### Step 4: Calculate the Amount Needed in the 6th Year

To find the remaining amount needed in the 6th year:

\[
\text{Remaining Amount} = 10,000 - 8,173.21 \approx 1,826.79
\]

The remaining amount will accumulate interest over the 6th year, so the last deposit should be:

\[
\text{Last Deposit} = \frac{1,826.79}{1.06} \approx 1,723.39
\]

### Final Answers:

- It will take **6 years** to reach the $10,000 goal.
- The last deposit will be **$1,723.39**.

Would you like more details or have any questions?

### Related Questions:
1. How would the required years change if the interest rate were 5% instead of 6%?
2. What if you made semi-annual deposits instead of annual ones? How would that affect the number of years?
3. How much would you need to deposit annually if you wanted to reach $10,000 in exactly 5 years?
4. If you increased your annual deposit by 3% each year, how many years would it take to reach $10,000?
5. What would be the impact of an initial deposit of $1,000 on the number of years required?

### Tip:
Always consider how interest rates and compounding frequency affect the time required to reach your financial goals.

Learn how to calculate the time required to accumulate $10,000 by making annual deposits of $1,450 with a 6% annual interest rate. Discover the final deposit amount and detailed step-by-step calculations. Suitable for advanced high school students or anyone interested in financial mathematics.

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