Suppose that you earned a bachelor's degree and now you're teaching high school. The school district offers teachers the opportunity to take a year off to earn a master's degree. To achieve this goal, you deposit $$ \$ 3000 $$ at the end of each year in an annuity that pays $$ 5.5 \% $$ compounded annually. a. How much will you have saved at the end of five years? b. Find the interest.

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To solve this problem, we will use the formula for the future value of an ordinary annuity, which is given by:

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

where:
- $$ FV $$ is the future value of the annuity,
- $$ P $$ is the annual payment (deposit),
- $$ r $$ is the annual interest rate (as a decimal),
- $$ n $$ is the number of years.

### Given:
- $$ P = 3000 $$
- $$ r = 5.5\% = 0.055 $$
- $$ n = 5 $$

### Step a: Calculate the future value at the end of five years.

Substituting the known values into the formula:

$$
FV = 3000 \times \frac{(1 + 0.055)^5 - 1}{0.055}
$$

Now, let's calculate this step by step.

1. Calculate $$ (1 + 0.055)^5 $$.
2. Subtract 1 from the result.
3. Divide by $$ 0.055 $$.
4. Multiply by $$ 3000 $$.

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{3000\left(\left(1+0.055\right)^{5}-1\right)}{0.055}$$
- step1: Add the numbers:
  $$\frac{3000\left(1.055^{5}-1\right)}{0.055}$$
- step2: Convert the expressions:
  $$\frac{3000\left(\left(\frac{211}{200}\right)^{5}-1\right)}{0.055}$$
- step3: Subtract the numbers:
  $$\frac{3000\times \frac{211^{5}-200^{5}}{200^{5}}}{0.055}$$
- step4: Multiply the numbers:
  $$\frac{\frac{3\times 211^{5}-3\times 200^{5}}{4096\times 5^{7}}}{0.055}$$
- step5: Convert the expressions:
  $$\frac{\frac{3\times 211^{5}-3\times 200^{5}}{4096\times 5^{7}}}{\frac{11}{200}}$$
- step6: Multiply by the reciprocal:
  $$\frac{3\times 211^{5}-3\times 200^{5}}{4096\times 5^{7}}\times \frac{200}{11}$$
- step7: Reduce the numbers:
  $$\frac{3\times 211^{5}-3\times 200^{5}}{512\times 5^{7}}\times \frac{25}{11}$$
- step8: Rewrite the expression:
  $$\frac{3\times 211^{5}-3\times 200^{5}}{512\times 5^{7}}\times \frac{5^{2}}{11}$$
- step9: Reduce the numbers:
  $$\frac{3\times 211^{5}-3\times 200^{5}}{512\times 5^{5}}\times \frac{1}{11}$$
- step10: Multiply the fractions:
  $$\frac{3\times 211^{5}-3\times 200^{5}}{1600000\times 11}$$
- step11: Multiply:
  $$\frac{3\times 211^{5}-3\times 200^{5}}{17600000}$$
The future value of the annuity at the end of five years is approximately:

$$
FV \approx 16743.27
$$

### Step b: Find the interest earned.

To find the interest earned, we need to subtract the total amount deposited from the future value.

1. **Total amount deposited** over 5 years:
   $$
   \text{Total Deposits} = P \times n = 3000 \times 5 = 15000
   $$

2. **Interest earned**:
   $$
   \text{Interest} = FV - \text{Total Deposits} = 16743.27 - 15000
   $$

Now, let's calculate the interest earned.
Calculate the value by following steps:
- step0: Calculate:
  $$16743.27-15000$$
- step1: Subtract the numbers:
  $$1743.27$$
The interest earned over the five years is approximately:

$$
\text{Interest} \approx 1743.27
$$

### Summary of Results:
- **Total amount saved at the end of five years**: $$ \approx 16743.27 $$
- **Total interest earned**: $$ \approx 1743.27 $$
At the end of five years, you will have saved approximately \$16,743.27, and the interest earned will be approximately \$1,743.27.

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