
Find the payment that should be used for the annuity due whose future value is given. Assume that the compounding period is the same as the payment period.
​$14 comma 000​; quarterly payments for 10 ​years; interest rate 6.1​%

To find the payment amount for an annuity due with a future value of $14,000, quarterly payments for 10 years, and an interest rate of 6.1%, we can use the **Future Value of an Annuity Due formula**:

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

where:
- $FV$ = future value of the annuity (given as $14,000)
- $P$ = payment amount per period (what we need to find)
- $i$ = interest rate per period
- $n$ = total number of payments

### Step 1: Determine the interest rate per period and the number of payments

1. Since the payments are quarterly, we divide the annual interest rate by 4:
   \[
   i = \frac{6.1\%}{4} = \frac{0.061}{4} = 0.01525
   \]

2. The total number of payments over 10 years with quarterly payments is:
   \[
   n = 10 \times 4 = 40
   \]

### Step 2: Substitute the values into the formula

Rearrange the formula to solve for $P$:
\[
P = \frac{FV}{\frac{(1 + i)^n - 1}{i} \cdot (1 + i)}
\]

Substitute $FV = 14,000$, $i = 0.01525$, and $n = 40$:
\[
P = \frac{14000}{\frac{(1 + 0.01525)^{40} - 1}{0.01525} \cdot (1 + 0.01525)}
\]

Now, let’s calculate it step-by-step.

1. Calculate $(1 + 0.01525)^{40}$.
2. Subtract 1 from the result.
3. Divide by 0.01525.
4. Multiply by $1 + 0.01525$.
5. Finally, divide 14,000 by the result from step 4 to find $P$.

### Step 3: Calculate $P$ (using a calculator or computational tool)

After calculating, we find:

\[
P \approx 251.91
\]

### Answer:
The quarterly payment for the annuity due should be approximately **$251.91**.

Would you like further details or have any questions?

---

### Related Questions

1. How would the payment amount change if the annuity was an ordinary annuity instead of an annuity due

Find the payment that should be used for the annuity due whose future value is given. Assume that the compounding period is the same as the payment period.
​$14,000; quarterly payments for 10 ​years; interest rate 6.1​%

Learn how to determine the quarterly payment for an annuity due with a future value of $14,000, an interest rate of 6.1%, and a 10-year duration. This problem uses the future value of annuity due formula, covering essential concepts in financial mathematics. Suitable for high school seniors or college finance students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation