Future value (with changing years). Dixie Bank offers a certificate of deposit with an option to select your own investment period. Jonathan has $$ \$ 7,500 $$ for his CD investment. If the bank is offering a $$ 5 \% $$ interest rate, compounded annually, how much will the CD be worth at maturity if Jonathan picks a a. three-year investment period? b. five-year investment period? c. ten-year investment period? d. twenty-year investment period?

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To calculate the future value of the certificate of deposit (CD) with a changing investment period, we can use the formula for compound interest:

$$ FV = PV \times (1 + r)^n $$

where:
- $$ FV $$ is the future value of the investment
- $$ PV $$ is the present value of the investment (initial amount)
- $$ r $$ is the annual interest rate
- $$ n $$ is the number of years the money is invested

Given:
- Initial amount (present value) = $7,500
- Annual interest rate = 5% = 0.05
- We need to calculate the future value for different investment periods:
  a. Three-year investment period
  b. Five-year investment period
  c. Ten-year investment period
  d. Twenty-year investment period

Let's calculate the future value for each investment period:
Calculate the value by following steps:
- step0: Calculate:
  $$7500\left(1+0.05\right)^{3}$$
- step1: Add the numbers:
  $$7500\times 1.05^{3}$$
- step2: Convert the expressions:
  $$7500\left(\frac{21}{20}\right)^{3}$$
- step3: Evaluate the power:
  $$7500\times \frac{9261}{8000}$$
- step4: Multiply:
  $$\frac{138915}{16}$$
a. For a three-year investment period, the future value of the CD will be $8682.1875.

Now, let's calculate the future value for the remaining investment periods:
Calculate the value by following steps:
- step0: Calculate:
  $$7500\left(1+0.05\right)^{5}$$
- step1: Add the numbers:
  $$7500\times 1.05^{5}$$
- step2: Convert the expressions:
  $$7500\left(\frac{21}{20}\right)^{5}$$
- step3: Simplify:
  $$7500\times \frac{21^{5}}{20^{5}}$$
- step4: Rewrite the expression:
  $$20\times 375\times \frac{21^{5}}{20^{5}}$$
- step5: Reduce the numbers:
  $$375\times \frac{21^{5}}{20^{4}}$$
- step6: Rewrite the expression:
  $$125\times 3\times \frac{21^{5}}{20^{4}}$$
- step7: Rewrite the expression:
  $$125\times 3\times \frac{21^{5}}{5^{4}\times 4^{4}}$$
- step8: Rewrite the expression:
  $$5^{3}\times 3\times \frac{21^{5}}{5^{4}\times 4^{4}}$$
- step9: Reduce the numbers:
  $$3\times \frac{21^{5}}{5\times 4^{4}}$$
- step10: Multiply:
  $$\frac{3\times 21^{5}}{1280}$$
b. For a five-year investment period, the future value of the CD will be $9572.11171875.

Now, let's calculate the future value for the remaining investment periods:
Calculate the value by following steps:
- step0: Calculate:
  $$7500\left(1+0.05\right)^{10}$$
- step1: Add the numbers:
  $$7500\times 1.05^{10}$$
- step2: Convert the expressions:
  $$7500\left(\frac{21}{20}\right)^{10}$$
- step3: Simplify:
  $$7500\times \frac{21^{10}}{20^{10}}$$
- step4: Rewrite the expression:
  $$20\times 375\times \frac{21^{10}}{20^{10}}$$
- step5: Reduce the numbers:
  $$375\times \frac{21^{10}}{20^{9}}$$
- step6: Rewrite the expression:
  $$125\times 3\times \frac{21^{10}}{20^{9}}$$
- step7: Rewrite the expression:
  $$125\times 3\times \frac{21^{10}}{5^{9}\times 4^{9}}$$
- step8: Rewrite the expression:
  $$5^{3}\times 3\times \frac{21^{10}}{5^{9}\times 4^{9}}$$
- step9: Reduce the numbers:
  $$3\times \frac{21^{10}}{5^{6}\times 4^{9}}$$
- step10: Multiply the fractions:
  $$\frac{3\times 21^{10}}{15625\times 4^{9}}$$
c. For a ten-year investment period, the future value of the CD will be approximately $12216.709701.

Now, let's calculate the future value for the remaining investment period:
Calculate the value by following steps:
- step0: Calculate:
  $$7500\left(1+0.05\right)^{20}$$
- step1: Add the numbers:
  $$7500\times 1.05^{20}$$
- step2: Convert the expressions:
  $$7500\left(\frac{21}{20}\right)^{20}$$
- step3: Simplify:
  $$7500\times \frac{21^{20}}{20^{20}}$$
- step4: Rewrite the expression:
  $$20\times 375\times \frac{21^{20}}{20^{20}}$$
- step5: Reduce the numbers:
  $$375\times \frac{21^{20}}{20^{19}}$$
- step6: Rewrite the expression:
  $$125\times 3\times \frac{21^{20}}{20^{19}}$$
- step7: Rewrite the expression:
  $$125\times 3\times \frac{21^{20}}{5^{19}\times 4^{19}}$$
- step8: Rewrite the expression:
  $$5^{3}\times 3\times \frac{21^{20}}{5^{19}\times 4^{19}}$$
- step9: Reduce the numbers:
  $$3\times \frac{21^{20}}{5^{16}\times 4^{19}}$$
- step10: Multiply the fractions:
  $$\frac{3\times 21^{20}}{5^{16}\times 4^{19}}$$
d. For a twenty-year investment period, the future value of the CD will be approximately $19899.732789.

Therefore, the future values of the CD for different investment periods are:
a. Three-year investment period: $8682.1875
b. Five-year investment period: $9572.11171875
c. Ten-year investment period: $12216.709701
d. Twenty-year investment period: $19899.732789
The future values of the CD for different investment periods are: a. Three years: $8,682.19 b. Five years: $9,572.11 c. Ten years: $12,216.71 d. Twenty years: $19,899.73

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