Challenge question. In the chapter text, we dealt exclusively with a single lump sum, but often we may be looking at several lump-sum values simultaneously. Let's consider the retirement plan of a couple. Currently, the couple has four different investments: a $$ 401(\mathrm{k}) $$ plan, two pension plans, and a personal portfolio. The couple is 6 years away from retirement. They believe they have sufficient money in their plans today so that they do not have to contribute to the plans over the next 6 years and will still meet their $$ \$ 1.5 $$ million retirement goal. Here are the current values and growth rates of their plans: $$ 401(\mathrm{k}): \$ 93,000 $$ growing at $$ 6 \% $$. Pension Plan One: $$ \$ 287,000 $$ growing at $$ 6.75 \% $$. Pension Plan Two: $$ \$ 250,000 $$ growing at $$ 7.5 \% $$. Personal Portfolio: $$ \$ 141,000 $$ growing at $$ 10 \% $$. What is the value of the Pension Plan Two 6 vears from now?

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To find the value of the Pension Plan Two 6 years from now, we can use the formula for compound interest:

$$ A = P(1 + r)^n $$

where:
- $$ A $$ is the future value of the investment
- $$ P $$ is the present value of the investment
- $$ r $$ is the annual growth rate
- $$ n $$ is the number of years

Given:
- Present value of Pension Plan Two: $$ P = \$250,000 $$
- Annual growth rate: $$ r = 7.5\% = 0.075 $$
- Number of years: $$ n = 6 $$

Substitute the values into the formula to find the future value of Pension Plan Two 6 years from now.
Calculate the value by following steps:
- step0: Calculate:
  $$250000\left(1+0.075\right)^{6}$$
- step1: Add the numbers:
  $$250000\times 1.075^{6}$$
- step2: Convert the expressions:
  $$250000\left(\frac{43}{40}\right)^{6}$$
- step3: Simplify:
  $$250000\times \frac{43^{6}}{40^{6}}$$
- step4: Rewrite the expression:
  $$40\times 6250\times \frac{43^{6}}{40^{6}}$$
- step5: Reduce the numbers:
  $$6250\times \frac{43^{6}}{40^{5}}$$
- step6: Rewrite the expression:
  $$10\times 625\times \frac{43^{6}}{40^{5}}$$
- step7: Rewrite the expression:
  $$10\times 625\times \frac{43^{6}}{10^{5}\times 4^{5}}$$
- step8: Reduce the numbers:
  $$625\times \frac{43^{6}}{10^{4}\times 4^{5}}$$
- step9: Rewrite the expression:
  $$625\times \frac{43^{6}}{5^{4}\times 2^{4}\times 4^{5}}$$
- step10: Rewrite the expression:
  $$5^{4}\times \frac{43^{6}}{5^{4}\times 2^{4}\times 4^{5}}$$
- step11: Reduce the numbers:
  $$1\times \frac{43^{6}}{2^{4}\times 4^{5}}$$
- step12: Multiply:
  $$\frac{43^{6}}{2^{14}}$$
- step13: Simplify:
  $$\frac{43^{6}}{16384}$$
The value of the Pension Plan Two 6 years from now is approximately $385,825.38.
The Pension Plan Two will be worth approximately \$385,825.38 six years from now.

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