Sure, let's solve this step-by-step using the compound interest formula.
The compound interest formula is given by:
[tex]\[ F = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( F \)[/tex] is the final amount
- [tex]\( P \)[/tex] is the principal amount (initial deposit)
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal)
- [tex]\( n \)[/tex] is the number of times interest is compounded per year
- [tex]\( t \)[/tex] is the number of years the money is invested or borrowed for
Given:
- [tex]\( P = 700 \)[/tex] dollars
- [tex]\( r = 0.105 \)[/tex] (10.5% interest rate expressed as a decimal)
- [tex]\( n = 12 \)[/tex] (interest is compounded monthly)
- [tex]\( t = 16 \)[/tex] years
Let's plug these values into the formula:
1. Calculate the term inside the parentheses:
[tex]\[ 1 + \frac{r}{n} = 1 + \frac{0.105}{12} \][/tex]
[tex]\[ = 1 + 0.00875 \][/tex]
[tex]\[ = 1.00875 \][/tex]
2. Raise this result to the power of [tex]\( nt \)[/tex]:
[tex]\[ (1.00875)^{12 \times 16} = (1.00875)^{192} \][/tex]
3. Multiply the principal amount [tex]\( P \)[/tex] by this result to find [tex]\( F \)[/tex]:
[tex]\[ F = 700 \times (1.00875)^{192} \][/tex]
After performing the calculations (considering the values), the final balance [tex]\( F \)[/tex] after 16 years is:
[tex]\[ F \approx 3728.54 \][/tex]
So, the balance after 16 years will be approximately \$3728.54.
