If $795 is invested in an account that earns 11.75%, compounded annually, what will the account balance be after 27 years?
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To find the account balance after 27 years, we need to use the formula for compound interest:
\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]
Where:
- A is the final amount (account balance),
- P is the principal amount (initial investment),
- r is the annual interest rate (expressed as a decimal),
- n is the number of times the interest is compounded per year, and
- t is the number of years.
In this case, the principal amount is $795, the annual interest rate is 11.75% (or 0.1175 as a decimal), the interest is compounded annually (so n = 1), and the number of years is 27.
Now, we can calculate the account balance:
\[A = 795 \left(1 + \frac{0.1175}{1}\right)^{(1)(27)}\]
Let's simplify this calculation step by step:
Step 1: Simplify the fraction inside the parentheses:
\[\frac{0.1175}{1} = 0.1175\]
Step 2: Raise the simplified fraction to the power of 27:
\[0.1175^{(1)(27)} = 0.1175^{27}\]
Step 3: Calculate the value of the exponent:
\[0.1175^{27} \approx 7.29624302011\] (rounded to 11 decimal places)
Step 4: Multiply the principal amount by the value obtained in step 3:
\[A = 795 \times 7.29624302011\]
Step 5: Calculate the final account balance:
\[A \approx 5796.28\] (rounded to 2 decimal places)
Therefore, the account balance after 27 years will be approximately $5796.28.