Jenny is planning for retirement. She deposits $825 at the end of each year in an ordinary annuity that pays 2%  interest, compounded annually. Answer each part. Do not round any intermediate computations nor answers.
(a)Find the total value of the annuity at the end of the 1st  year.
(b)Find the total value of the annuity at the end of the 2nd  year.
(c)Find the total value of the annuity at the end of the 3rd  year.

To solve the problem, we need to determine the total value of the annuity at the end of each year for Jenny's retirement plan. This involves calculating the future value of an ordinary annuity, where the payment is made at the end of each year and interest is compounded annually.

The formula to calculate the future value of an ordinary annuity is:

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

Where:
- $ FV $ is the future value of the annuity.
- $ P $ is the payment per period (in this case, $825).
- $ r $ is the interest rate per period (2% or 0.02 annually).
- $ n $ is the number of periods.

### (a) Total value of the annuity at the end of the 1st year

At the end of the 1st year, only one payment of $825 has been made, and it earns interest at 2%:

\[
FV_1 = 825 \times (1 + 0.02) = 825 \times 1.02 = 841.50
\]

So, the total value at the end of the 1st year is **$841.50**.

### (b) Total value of the annuity at the end of the 2nd year

At the end of the 2nd year:
- The first payment of $825 has earned interest for two years.
- The second payment of $825 has earned interest for one year.

The future value is calculated by adding these:

\[
FV_2 = 825 \times (1.02)^2 + 825 \times 1.02 = 825 \times 1.0404 + 825 \times 1.02 = 858.33 + 841.50 = 1699.83
\]

So, the total value at the end of the 2nd year is **$1699.83**.

### (c) Total value of the annuity at the end of the 3rd year

At the end of the 3rd year:
- The first payment of $825 has earned interest for three years.
- The second payment of $825 has earned interest for two years.
- The third payment of $825 has earned interest for one year.

The future value is:

\[
FV_3 = 825 \times (1.02)^3 + 825 \times (1.02)^2 + 825 \times 1.02 
\]
\[
FV_3 = 825 \times 1.061208 + 825 \times 1.0404 + 825 \times 1.02 = 876.4986 + 858.33 + 841.50 = 2576.3286
\]

So, the total value at the end of the 3rd year is **$2576.33**.

### Summary of Results:
- **(a)** At the end of the 1st year: **$841.50**
- **(b)** At the end of the 2nd year: **$1699.83**
- **(c)** At the end of the 3rd year: **$2576.33**

Would you like any further details or have any other questions? 

Here are five related questions to expand your understanding:

1. How would the total value change if the interest rate were 3% instead of 2%?
2. What would be the total value at the end of 5 years?
3. How does the compounding frequency affect the future value of the annuity?
4. How much more would Jenny have if she deposited $900 annually instead of $825?
5. What would happen if Jenny made deposits at the beginning of each year (annuity due) instead of at the end?

**Tip:** In retirement planning, it's crucial to start saving early because the power of compounding increases the value of your investments significantly over time.

Learn how to calculate the future value of an ordinary annuity with compound interest. Solve Jenny's retirement plan problem step-by-step. Includes detailed computations for each year and explanations of compound interest principles.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation