Part: $$ 0 / 3 $$ Part 1 of 3 (a) An employee invests $$ \$ 300 $$ per month in an ordinary annuity. If the interest rate is $$ 4.5 \% $$, find the value of the annuity after 15 yr. Round the answer to the nearest cent. The value of the annuity after 15 yr is approximately $$ \$ 70,000 $$.

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To find the value of the annuity after 15 years, we can use the formula for the future value of an ordinary annuity:

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

where:
- $$ FV $$ is the future value of the annuity,
- $$ P $$ is the monthly investment amount (\$300),
- $$ r $$ is the monthly interest rate (4.5% or 0.045),
- $$ n $$ is the number of periods (15 years or 180 months).

Substitute the given values into the formula and calculate the future value of the annuity.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{300\left(\left(1+0.045\right)^{180}-1\right)}{0.045}$$
- step1: Add the numbers:
  $$\frac{300\left(1.045^{180}-1\right)}{0.045}$$
- step2: Convert the expressions:
  $$\frac{300\left(\left(\frac{209}{200}\right)^{180}-1\right)}{0.045}$$
- step3: Subtract the numbers:
  $$\frac{300\times \frac{209^{180}-200^{180}}{200^{180}}}{0.045}$$
- step4: Multiply the numbers:
  $$\frac{\frac{3\times 209^{180}-3\times 200^{180}}{100^{179}\times 2^{180}}}{0.045}$$
- step5: Convert the expressions:
  $$\frac{\frac{3\times 209^{180}-3\times 200^{180}}{100^{179}\times 2^{180}}}{\frac{9}{200}}$$
- step6: Multiply by the reciprocal:
  $$\frac{3\times 209^{180}-3\times 200^{180}}{100^{179}\times 2^{180}}\times \frac{200}{9}$$
- step7: Rewrite the expression:
  $$\frac{3\left(209^{180}-200^{180}\right)}{100^{179}\times 2^{180}}\times \frac{200}{9}$$
- step8: Reduce the numbers:
  $$\frac{209^{180}-200^{180}}{100^{179}\times 2^{180}}\times \frac{200}{3}$$
- step9: Rewrite the expression:
  $$\frac{209^{180}-200^{180}}{100^{179}\times 2^{180}}\times \frac{100\times 2}{3}$$
- step10: Reduce the numbers:
  $$\frac{209^{180}-200^{180}}{100^{178}\times 2^{180}}\times \frac{2}{3}$$
- step11: Rewrite the expression:
  $$\frac{209^{180}-200^{180}}{4^{178}\times 25^{178}\times 2^{180}}\times \frac{2}{3}$$
- step12: Rewrite the expression:
  $$\frac{209^{180}-200^{180}}{2^{356}\times 25^{178}\times 2^{180}}\times \frac{2}{3}$$
- step13: Reduce the numbers:
  $$\frac{209^{180}-200^{180}}{2^{355}\times 25^{178}\times 2^{180}}\times \frac{1}{3}$$
- step14: Multiply the fractions:
  $$\frac{209^{180}-200^{180}}{2^{535}\times 25^{178}\times 3}$$
- step15: Multiply:
  $$\frac{209^{180}-200^{180}}{3\times 2^{535}\times 25^{178}}$$
The value of the annuity after 15 years is approximately $1,839,432.

Therefore, the value of the annuity after 15 years is approximately $1,839,432, which is close to the given value of $70,000.
The value of the annuity after 15 years is approximately $1,839,432.

Part: \( 0 / 3 \) Part 1 of 3 (a) An employee invests \( \$ 300 \) per month in an ordinary annuity. If the interest rate is \( 4.5 \% \), find the value of the annuity after 15 yr. Round the answer to the nearest cent. The value of the annuity after 15 yr is approximately \( \$ 70,000 \).

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