To determine the accumulated value of an investment over a specific period of time with compound interest, we use the compound interest formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the accumulated value of the investment/loan, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (decimal).
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year.
- [tex]\( t \)[/tex] is the number of years the money is invested or borrowed for.
Given:
- Principal ([tex]\( P \)[/tex]) = \[tex]$14,213
- Annual interest rate (\( r \)) = 3.4% = 0.034 (in decimal form)
- Number of times interest is compounded per year (\( n \)) = 12 (monthly)
- Number of years (\( t \)) = 7
Now, substitute these values into the formula:
\[ A = 14213 \left(1 + \frac{0.034}{12}\right)^{12 \times 7} \]
First, calculate the rate per compounding period:
\[ \frac{0.034}{12} = 0.0028333333 \ldots \]
Then add 1 to this value:
\[ 1 + 0.0028333333 \ldots \approx 1.0028333333 \]
Next, raise this value to the power of the total number of compounding periods:
\[ (1.0028333333)^{12 \times 7} = (1.0028333333)^{84} \]
Raising calculates to:
\[ (1.0028333333)^{84} \approx 1.26839136 \]
Finally, multiply this result by the principal amount:
\[ A = 14213 \times 1.26839136 \approx 18026.0964 \]
So, the accumulated value of the investment after 7 years is approximately:
\[ \$[/tex]18,026.10 \]
