a. Use the appropriate formula to determine the periodic deposit.
b. How much of the financial goal comes from deposits and how much comes from interest?
Periodic Deposit
$? at the end of each year
Rate
6% compounded annually
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Time
17 years
Financial Goal
$120,000
HW Score: 24.96%, 7
Points: 0 of 2
a. The periodic deposit is
(Do not round until the final answer. Then round up to the nearest dollar.)

To find the periodic deposit, we'll use the **future value of an annuity formula**, which is appropriate for problems where regular deposits are made into an account that earns interest.

The formula for the future value of an annuity compounded at the end of each period is:

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

Where:
- $ FV $ is the future value (financial goal),
- $ P $ is the periodic deposit,
- $ r $ is the interest rate per period (in decimal form),
- $ t $ is the number of periods.

### Given:
- Financial goal $ FV = 120,000 $,
- Interest rate $ r = 6\% = 0.06 $ (compounded annually),
- Time $ t = 17 $ years.

We need to find the periodic deposit $ P $. Rearranging the formula to solve for $ P $:

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

### Step 1: Calculate the future value factor

\[
(1 + r)^t - 1 = (1 + 0.06)^{17} - 1 = (1.06)^{17} - 1
\]

\[
(1.06)^{17} \approx 2.71264
\]

\[
2.71264 - 1 = 1.71264
\]

### Step 2: Divide by the interest rate

\[
\frac{1.71264}{0.06} = 28.544
\]

### Step 3: Calculate the periodic deposit

\[
P = \frac{120,000}{28.544} \approx 4,203.23
\]

Since we round up to the nearest dollar, the periodic deposit is:

\[
\boxed{4,204}
\]

### Part (b): How much comes from deposits and how much comes from interest?

The total amount of money deposited is $ 4,204 \times 17 = 71,468 $.

The amount from interest is:

\[
120,000 - 71,468 = 48,532
\]

So, $71,468 comes from deposits, and $48,532 comes from interest.

---

Would you like more details or have any questions?

Here are 5 related questions:
1. How does increasing the interest rate affect the periodic deposit?
2. What if the deposits were made monthly instead of annually?
3. How does reducing the time period affect the total interest earned?
4. What happens if the financial goal is increased to $150,000?
5. How is this formula different when compounded more frequently (e.g., monthly)?

**Tip:** When solving for periodic deposits in annuities, always double-check whether deposits are made at the beginning or the end of the period, as the formula can change accordingly.

a. Use the appropriate formula to determine the periodic deposit.
b. How much of the financial goal comes from deposits and how much comes from interest?
Periodic Deposit: $? at the end of each year
Rate: 6% compounded annually
Time: 17 years
Financial Goal: $120,000

Learn how to calculate the periodic deposit needed to achieve a financial goal of $120,000 in 17 years with 6% annual compound interest. This problem covers key financial concepts, including the future value of an annuity formula and interest contributions. Suitable for students in Grades 10-12.

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