Angela deposits $$ \$ 3500 $$ into an account with an APR of $$ 2.5 \% $$ for 11 years. Find the future value of the account if interest is compounded monthly. Round your answer to the nearest hundredth, if necessary.

Question

UpStudy AI · Accepted Answer

To find the future value of an account with compound interest, we can use the formula:

$$
FV = P \left(1 + \frac{r}{n}\right)^{nt}
$$

where:
- $$ FV $$ is the future value of the investment/loan, including interest,
- $$ P $$ is the principal investment amount (the initial deposit or loan amount),
- $$ r $$ is the annual interest rate (decimal),
- $$ n $$ is the number of times that interest is compounded per unit $$ t $$,
- $$ t $$ is the time the money is invested or borrowed for, in years.

Given:
- $$ P = 3500 $$
- $$ r = 2.5\% = 0.025 $$
- $$ n = 12 $$ (since the interest is compounded monthly)
- $$ t = 11 $$

Now, we can substitute these values into the formula:

$$
FV = 3500 \left(1 + \frac{0.025}{12}\right)^{12 \times 11}
$$

Let's calculate this step by step.

1. Calculate $$ \frac{r}{n} $$:
   $$
   \frac{0.025}{12}
   $$

2. Calculate $$ nt $$:
   $$
   12 \times 11
   $$

3. Substitute these values into the formula and calculate $$ FV $$.

Now, I will perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$12\times 11$$
- step1: Multiply the numbers:
  $$132$$
Calculate or simplify the expression $$ 0.025/12 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.025}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{1}{40}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{1}{40}\times \frac{1}{12}$$
- step3: Multiply the fractions:
  $$\frac{1}{40\times 12}$$
- step4: Multiply:
  $$\frac{1}{480}$$
Calculate or simplify the expression $$ 3500*(1+0.00208333)^(132) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$3500\left(1+0.00208333\right)^{132}$$
- step1: Add the numbers:
  $$3500\times 1.00208333^{132}$$
- step2: Convert the expressions:
  $$3500\left(\frac{100208333}{100000000}\right)^{132}$$
- step3: Simplify:
  $$3500\times \frac{100208333^{132}}{100000000^{132}}$$
- step4: Rewrite the expression:
  $$500\times 7\times \frac{100208333^{132}}{100000000^{132}}$$
- step5: Rewrite the expression:
  $$500\times 7\times \frac{100208333^{132}}{250000^{132}\times 400^{132}}$$
- step6: Rewrite the expression:
  $$500\times 7\times \frac{100208333^{132}}{500^{264}\times 400^{132}}$$
- step7: Reduce the numbers:
  $$7\times \frac{100208333^{132}}{500^{263}\times 400^{132}}$$
- step8: Multiply the fractions:
  $$\frac{7\times 100208333^{132}}{500^{263}\times 400^{132}}$$
Now that we have the necessary calculations:

1. The monthly interest rate $$ \frac{r}{n} $$ is approximately $$ 0.00208333 $$.
2. The total number of compounding periods $$ nt $$ is $$ 132 $$.

Now we can substitute these values into the future value formula:

$$
FV = 3500 \left(1 + 0.00208333\right)^{132}
$$

Calculating this gives us:

$$
FV \approx 3500 \times (1.00208333)^{132} \approx 4606.54
$$

Thus, the future value of the account after 11 years, rounded to the nearest hundredth, is approximately:

$$
\boxed{4606.54}
$$
The future value of Angela's account after 11 years is approximately \$4,606.54.

Angela deposits into an account with an APR of for 11 years. Find the future value of the account if interest is compounded monthly. Round your
answer to the nearest hundredth, if necessary.

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Angela deposits into an account with an APR of for 11 years. Find the future value of the account if interest is compounded monthly. Round your answer to the nearest hundredth, if necessary.