Suppose you invest $$ \$ 170 $$ a month for 6 years into an account earning $$ 8 \% $$ compounded monthly. After 6 years, you leave the money, without making additional deposits, in the account for another 24 years. How much will you have in the end? $$ \$ 93791.36 $$ Check Answer

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UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$170\left(\frac{\left(\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 6}-1\right)}{\left(\frac{0.08}{12}\right)}\right)$$
- step1: Remove the parentheses:
  $$170\left(\frac{\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)$$
- step2: Divide the terms:
  $$170\left(\frac{\left(1+\frac{1}{150}\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)$$
- step3: Add the numbers:
  $$170\left(\frac{\left(\frac{151}{150}\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)$$
- step4: Multiply the numbers:
  $$170\left(\frac{\left(\frac{151}{150}\right)^{72}-1}{\frac{0.08}{12}}\right)$$
- step5: Divide the terms:
  $$170\left(\frac{\left(\frac{151}{150}\right)^{72}-1}{\frac{1}{150}}\right)$$
- step6: Subtract the numbers:
  $$170\left(\frac{\frac{151^{72}-150^{72}}{150^{72}}}{\frac{1}{150}}\right)$$
- step7: Divide the terms:
  $$170\times \frac{151^{72}-150^{72}}{150^{71}}$$
- step8: Rewrite the expression:
  $$10\times 17\times \frac{151^{72}-150^{72}}{150^{71}}$$
- step9: Rewrite the expression:
  $$10\times 17\times \frac{151^{72}-150^{72}}{10^{71}\times 15^{71}}$$
- step10: Reduce the numbers:
  $$17\times \frac{151^{72}-150^{72}}{10^{70}\times 15^{71}}$$
- step11: Multiply the fractions:
  $$\frac{17\left(151^{72}-150^{72}\right)}{10^{70}\times 15^{71}}$$
- step12: Multiply:
  $$\frac{17\times 151^{72}-17\times 150^{72}}{10^{70}\times 15^{71}}$$
Calculate or simplify the expression $$ (170 * (((1 + (0.08 / 12))^(12 * 6) - 1) / (0.08 / 12))) * (1 + (0.08 / 12))^(12 * 24) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(170\left(\frac{\left(\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 6}-1\right)}{\left(\frac{0.08}{12}\right)}\right)\right)\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 24}$$
- step1: Remove the parentheses:
  $$170\left(\frac{\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 24}$$
- step2: Divide the terms:
  $$170\left(\frac{\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)\left(1+\frac{1}{150}\right)^{12\times 24}$$
- step3: Add the numbers:
  $$170\left(\frac{\left(1+\left(\frac{0.08}{12}\right)\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)\left(\frac{151}{150}\right)^{12\times 24}$$
- step4: Divide the terms:
  $$170\left(\frac{\left(1+\frac{1}{150}\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)\left(\frac{151}{150}\right)^{12\times 24}$$
- step5: Add the numbers:
  $$170\left(\frac{\left(\frac{151}{150}\right)^{12\times 6}-1}{\frac{0.08}{12}}\right)\left(\frac{151}{150}\right)^{12\times 24}$$
- step6: Multiply the numbers:
  $$170\left(\frac{\left(\frac{151}{150}\right)^{72}-1}{\frac{0.08}{12}}\right)\left(\frac{151}{150}\right)^{12\times 24}$$
- step7: Divide the terms:
  $$170\left(\frac{\left(\frac{151}{150}\right)^{72}-1}{\frac{1}{150}}\right)\left(\frac{151}{150}\right)^{12\times 24}$$
- step8: Subtract the numbers:
  $$170\left(\frac{\frac{151^{72}-150^{72}}{150^{72}}}{\frac{1}{150}}\right)\left(\frac{151}{150}\right)^{12\times 24}$$
- step9: Divide the terms:
  $$170\times \frac{151^{72}-150^{72}}{150^{71}}\times \left(\frac{151}{150}\right)^{12\times 24}$$
- step10: Multiply the numbers:
  $$170\times \frac{151^{72}-150^{72}}{150^{71}}\times \left(\frac{151}{150}\right)^{288}$$
- step11: Multiply the terms:
  $$\frac{17\times 151^{72}-17\times 150^{72}}{10^{70}\times 15^{71}}\times \left(\frac{151}{150}\right)^{288}$$
- step12: Evaluate the power:
  $$\frac{17\times 151^{72}-17\times 150^{72}}{10^{70}\times 15^{71}}\times \frac{151^{288}}{150^{288}}$$
- step13: Multiply the fractions:
  $$\frac{\left(17\times 151^{72}-17\times 150^{72}\right)\times 151^{288}}{10^{70}\times 15^{71}\times 150^{288}}$$
- step14: Multiply:
  $$\frac{17\times 151^{360}-17\times 150^{72}\times 151^{288}}{10^{70}\times 15^{71}\times 150^{288}}$$
To determine how much you will have in the end after investing $$ \$170 $$ a month for 6 years into an account earning $$ 8\% $$ compounded monthly, we can break the problem into two parts:

1. **Calculate the future value of the monthly investments after 6 years.**
2. **Calculate the future value of that amount after leaving it for another 24 years without additional deposits.**

### Step 1: Future Value of Monthly Investments

The formula for the future value of a series of monthly investments (ordinary annuity) is given by:

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

Where:
- $$ P $$ = monthly investment (\$170)
- $$ r $$ = monthly interest rate (annual rate / 12)
- $$ n $$ = total number of investments (months)

Given:
- Annual interest rate = $$ 8\% $$ or $$ 0.08 $$
- Monthly interest rate $$ r = \frac{0.08}{12} $$
- Total months $$ n = 6 \times 12 = 72 $$

Now, substituting the values into the formula:

$$
FV = 170 \times \frac{(1 + \frac{0.08}{12})^{72} - 1}{\frac{0.08}{12}}
$$

Calculating this gives:

$$
FV \approx 15644.31
$$

### Step 2: Future Value After 24 More Years

Now, we take the amount from Step 1 and let it grow for another 24 years (or 288 months) without additional deposits. The formula for the future value of a single sum is:

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

Where:
- $$ PV $$ = present value from Step 1
- $$ r $$ = monthly interest rate
- $$ n $$ = total number of months (24 years = 288 months)

Substituting the values:

$$
FV = 15644.31 \times (1 + \frac{0.08}{12})^{288}
$$

Calculating this gives:

$$
FV \approx 106031.40
$$

### Conclusion

After performing the calculations, the final amount in the account after 30 years (6 years of deposits and 24 years of growth) is approximately $$ \$106031.40 $$.

Thus, the answer $$ \$93791.36 $$ is incorrect. The correct final amount is approximately $$ \$106031.40 $$.
After 6 years of investing \$170 monthly at 8% compounded monthly, the investment grows to approximately \$106,031.40 over the next 24 years without additional deposits.

Suppose you invest \( \$ 170 \) a month for 6 years into an account earning \( 8 \% \) compounded monthly. After 6 years, you leave the money, without making additional deposits, in the account for another 24 years. How much will you have in the end? \( \$ 93791.36 \) Check Answer

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