8. Sidney opens the following investment portfolio: - A $$ \$ 10,000 $$ fund that earns $$ 5.45 \% $$, compounded annually - Monthly deposits of $$ \$ 300 $$ into an account earning $$ 4 \% $$, compounded monthly What will be the portfolio's value in 30 years when Sidney is ready to retire?

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To find the total value of Sidney's investment portfolio in 30 years, we need to calculate the future value of both components of the portfolio separately and then sum them up.

### Step 1: Calculate the future value of the initial investment

The formula for the future value of a single sum compounded annually is given by:

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

Where:
- $$ FV $$ = future value
- $$ P $$ = principal amount (initial investment)
- $$ r $$ = annual interest rate (as a decimal)
- $$ n $$ = number of years

For Sidney's initial investment:
- $$ P = 10000 $$
- $$ r = 0.0545 $$
- $$ n = 30 $$

Now, we can calculate the future value of the initial investment.

### Step 2: Calculate the future value of the monthly deposits

The formula for the future value of a series of regular deposits (an annuity) compounded at regular intervals is given by:

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

Where:
- $$ FV $$ = future value of the annuity
- $$ PMT $$ = amount of each payment
- $$ r $$ = annual interest rate (as a decimal)
- $$ n $$ = number of compounding periods per year
- $$ t $$ = number of years

For Sidney's monthly deposits:
- $$ PMT = 300 $$
- $$ r = 0.04 $$
- $$ n = 12 $$ (monthly deposits)
- $$ t = 30 $$

Now, we can calculate the future value of the monthly deposits.

Let's perform the calculations for both components.
Calculate the value by following steps:
- step0: Calculate:
  $$10000\left(1+0.0545\right)^{30}$$
- step1: Add the numbers:
  $$10000\times 1.0545^{30}$$
- step2: Convert the expressions:
  $$10000\left(\frac{2109}{2000}\right)^{30}$$
- step3: Simplify:
  $$10000\times \frac{2109^{30}}{2000^{30}}$$
- step4: Rewrite the expression:
  $$2000\times 5\times \frac{2109^{30}}{2000^{30}}$$
- step5: Reduce the numbers:
  $$5\times \frac{2109^{30}}{2000^{29}}$$
- step6: Rewrite the expression:
  $$5\times \frac{2109^{30}}{125^{29}\times 16^{29}}$$
- step7: Rewrite the expression:
  $$5\times \frac{2109^{30}}{5^{87}\times 16^{29}}$$
- step8: Reduce the numbers:
  $$1\times \frac{2109^{30}}{5^{86}\times 16^{29}}$$
- step9: Multiply the fractions:
  $$\frac{2109^{30}}{5^{86}\times 16^{29}}$$
Calculate or simplify the expression $$ 300*((1 + 0.04/12)^(12*30) - 1)/(0.04/12) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{300\left(\left(1+\frac{0.04}{12}\right)^{12\times 30}-1\right)}{\left(\frac{0.04}{12}\right)}$$
- step1: Remove the parentheses:
  $$\frac{300\left(\left(1+\frac{0.04}{12}\right)^{12\times 30}-1\right)}{\frac{0.04}{12}}$$
- step2: Divide the terms:
  $$\frac{300\left(\left(1+\frac{1}{300}\right)^{12\times 30}-1\right)}{\frac{0.04}{12}}$$
- step3: Add the numbers:
  $$\frac{300\left(\left(\frac{301}{300}\right)^{12\times 30}-1\right)}{\frac{0.04}{12}}$$
- step4: Multiply the numbers:
  $$\frac{300\left(\left(\frac{301}{300}\right)^{360}-1\right)}{\frac{0.04}{12}}$$
- step5: Subtract the numbers:
  $$\frac{300\times \frac{301^{360}-300^{360}}{300^{360}}}{\frac{0.04}{12}}$$
- step6: Divide the terms:
  $$\frac{300\times \frac{301^{360}-300^{360}}{300^{360}}}{\frac{1}{300}}$$
- step7: Multiply the numbers:
  $$\frac{\frac{301^{360}-300^{360}}{300^{359}}}{\frac{1}{300}}$$
- step8: Multiply by the reciprocal:
  $$\frac{301^{360}-300^{360}}{300^{359}}\times 300$$
- step9: Reduce the numbers:
  $$\frac{301^{360}-300^{360}}{300^{358}}\times 1$$
- step10: Multiply:
  $$\frac{301^{360}-300^{360}}{300^{358}}$$
### Step 1: Future Value of the Initial Investment

The future value of Sidney's initial investment of $$ \$10,000 $$ compounded annually at $$ 5.45\% $$ for 30 years is approximately:

$$
FV_{\text{initial}} \approx 49135.74
$$

### Step 2: Future Value of the Monthly Deposits

The future value of Sidney's monthly deposits of $$ \$300 $$ compounded monthly at $$ 4\% $$ for 30 years is approximately:

$$
FV_{\text{deposits}} \approx 208214.82
$$

### Step 3: Total Portfolio Value

Now, we can find the total value of the portfolio by summing the future values of both components:

$$
FV_{\text{total}} = FV_{\text{initial}} + FV_{\text{deposits}}
$$

Calculating this gives:

$$
FV_{\text{total}} \approx 49135.74 + 208214.82
$$

Let's calculate the total value.
Calculate the value by following steps:
- step0: Calculate:
  $$49135.7424+208214.821315$$
- step1: Add the numbers:
  $$257350.563715$$
The total value of Sidney's investment portfolio after 30 years is approximately:

$$
FV_{\text{total}} \approx 257350.56
$$

### Final Answer

Sidney's portfolio will be worth approximately **\$257,350.56** when he is ready to retire.
Sidney's portfolio will be worth approximately \$257,350.56 in 30 years.

8. Sidney opens the following investment portfolio: - A \( \$ 10,000 \) fund that earns \( 5.45 \% \), compounded annually - Monthly deposits of \( \$ 300 \) into an account earning \( 4 \% \), compounded monthly What will be the portfolio's value in 30 years when Sidney is ready to retire?

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