Suppose you invest $$ \$ 130 $$ a month for 4 years into an account earning $$ 8 \% $$ compounded monthly. After 4 years, you leave the money, without making additional deposits, in the account for another 29 years. How much will you have in the end? Question Help: Message instructor Calculator

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UpStudy AI · Accepted Answer

To solve this problem, we will break it down into two parts:

1. **Calculate the future value of the monthly investments after 4 years.**
2. **Calculate the future value of that amount after an additional 29 years without additional deposits.**

### Step 1: Future Value of Monthly Investments

We will use the future value of an annuity formula for the first part, which is given by:

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

Where:
- $$ FV $$ = future value of the annuity
- $$ P $$ = monthly deposit (\$130)
- $$ r $$ = monthly interest rate (annual rate / 12)
- $$ n $$ = total number of deposits (months)

Given:
- Monthly deposit $$ P = 130 $$
- Annual interest rate = $$ 8\% $$ or $$ 0.08 $$
- Monthly interest rate $$ r = \frac{0.08}{12} $$
- Total number of deposits $$ n = 4 \times 12 = 48 $$

Let's calculate $$ r $$ and then $$ FV $$ for the first 4 years.

### Step 2: Future Value After 29 Years

After calculating the future value from the first part, we will treat that amount as a single investment for the next 29 years. The formula for future value of a single sum is:

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

Where:
- $$ PV $$ = present value (the amount from the first part)
- $$ r $$ = monthly interest rate
- $$ t $$ = total time in months (29 years)

Now, let's perform the calculations.

1. Calculate $$ r $$ and $$ FV $$ for the first 4 years.
2. Use the result as $$ PV $$ to calculate the future value after 29 years.

Let's proceed with the calculations.
Simplify the expression by following steps:
- step0: Solution:
  $$PV\left(1+\left(\frac{0.08}{12}\right)\right)^{29\times 12}$$
- step1: Divide the terms:
  $$PV\left(1+\frac{1}{150}\right)^{29\times 12}$$
- step2: Add the numbers:
  $$PV\left(\frac{151}{150}\right)^{29\times 12}$$
- step3: Multiply the numbers:
  $$PV\left(\frac{151}{150}\right)^{348}$$
- step4: Reorder the terms:
  $$\left(\frac{151}{150}\right)^{348}PV$$
- step5: Simplify:
  $$\frac{151^{348}}{150^{348}}\times PV$$
Calculate or simplify the expression $$ 130 * ((1 + (0.08 / 12))^(4 * 12) - 1) / (0.08 / 12) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{130\left(\left(1+\left(\frac{0.08}{12}\right)\right)^{4\times 12}-1\right)}{\left(\frac{0.08}{12}\right)}$$
- step1: Remove the parentheses:
  $$\frac{130\left(\left(1+\left(\frac{0.08}{12}\right)\right)^{4\times 12}-1\right)}{\frac{0.08}{12}}$$
- step2: Divide the terms:
  $$\frac{130\left(\left(1+\frac{1}{150}\right)^{4\times 12}-1\right)}{\frac{0.08}{12}}$$
- step3: Add the numbers:
  $$\frac{130\left(\left(\frac{151}{150}\right)^{4\times 12}-1\right)}{\frac{0.08}{12}}$$
- step4: Multiply the numbers:
  $$\frac{130\left(\left(\frac{151}{150}\right)^{48}-1\right)}{\frac{0.08}{12}}$$
- step5: Subtract the numbers:
  $$\frac{130\times \frac{151^{48}-150^{48}}{150^{48}}}{\frac{0.08}{12}}$$
- step6: Divide the terms:
  $$\frac{130\times \frac{151^{48}-150^{48}}{150^{48}}}{\frac{1}{150}}$$
- step7: Multiply the numbers:
  $$\frac{\frac{13\times 151^{48}-13\times 150^{48}}{10^{47}\times 15^{48}}}{\frac{1}{150}}$$
- step8: Multiply by the reciprocal:
  $$\frac{13\times 151^{48}-13\times 150^{48}}{10^{47}\times 15^{48}}\times 150$$
- step9: Rewrite the expression:
  $$\frac{13\times 151^{48}-13\times 150^{48}}{10^{47}\times 15^{48}}\times 10\times 15$$
- step10: Reduce the numbers:
  $$\frac{13\times 151^{48}-13\times 150^{48}}{10^{46}\times 15^{48}}\times 15$$
- step11: Rewrite the expression:
  $$\frac{13\times 151^{48}-13\times 150^{48}}{10^{46}\times 15^{48}}\times 5\times 3$$
- step12: Rewrite the expression:
  $$\frac{13\times 151^{48}-13\times 150^{48}}{5^{46}\times 2^{46}\times 15^{48}}\times 5\times 3$$
- step13: Reduce the numbers:
  $$\frac{13\times 151^{48}-13\times 150^{48}}{5^{45}\times 2^{46}\times 15^{48}}\times 3$$
- step14: Rewrite the expression:
  $$\frac{13\times 151^{48}-13\times 150^{48}}{5^{45}\times 2^{46}\times 3^{48}\times 5^{48}}\times 3$$
- step15: Reduce the numbers:
  $$\frac{13\times 151^{48}-13\times 150^{48}}{5^{45}\times 2^{46}\times 3^{47}\times 5^{48}}\times 1$$
- step16: Multiply the fractions:
  $$\frac{13\times 151^{48}-13\times 150^{48}}{5^{93}\times 2^{46}\times 3^{47}}$$
Calculate or simplify the expression $$ 7325.488958 * (1 + (0.08 / 12))^(29 * 12) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$7325.488958\left(1+\left(\frac{0.08}{12}\right)\right)^{29\times 12}$$
- step1: Divide the terms:
  $$7325.488958\left(1+\frac{1}{150}\right)^{29\times 12}$$
- step2: Add the numbers:
  $$7325.488958\left(\frac{151}{150}\right)^{29\times 12}$$
- step3: Multiply the numbers:
  $$7325.488958\left(\frac{151}{150}\right)^{348}$$
- step4: Convert the expressions:
  $$\frac{3662744479}{500000}\left(\frac{151}{150}\right)^{348}$$
- step5: Multiply the numbers:
  $$\frac{3662744479\times 151^{348}}{500000\times 150^{348}}$$
### Step 1: Future Value of Monthly Investments

After calculating, the future value of the monthly investments after 4 years is approximately:

$$
FV_1 \approx 7325.49
$$

### Step 2: Future Value After 29 Years

Now, we will use this amount as the present value for the next 29 years. The future value after 29 years is approximately:

$$
FV_2 \approx 73970.09
$$

### Final Result

After a total of 33 years (4 years of monthly investments and 29 years of compounding), the total amount in the account will be approximately:

$$
\text{Total Amount} \approx 73970.09
$$

Thus, you will have approximately **$73,970.09** in the account at the end of the 33 years.
After 33 years, you will have approximately \$73,970.09 in the account.

Suppose you invest \( \$ 130 \) a month for 4 years into an account earning \( 8 \% \) compounded monthly. After 4 years, you leave the money, without making additional deposits, in the account for another 29 years. How much will you have in the end? Question Help: Message instructor Calculator

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