1. If Alexhas a loan balance of 12,000 that is compounded daily at a rate of 6.25% If he makes no payments on the account, what will be in 5 years?

To solve this problem, we need to use the formula for compound interest:<br /><br />\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]<br /><br />Where:<br />- \( A \) is the amount of money accumulated after \( n \) years, including interest.<br />- \( P \) is the principal amount (the initial amount of money).<br />- \( r \) is the annual interest rate (decimal).<br />- \( n \) is the number of times that interest is compounded per year.<br />- \( t \) is the time the money is invested for in years.<br /><br />Given:<br />- \( P = \$12,000 \)<br />- \( r = 6.25\% = 0.0625 \)<br />- \( n = 365 \) (since the interest is compounded daily)<br />- \( t = 5 \) years<br /><br />Now, plug these values into the formula:<br /><br />\[ A = 12000 \left(1 + \frac{0.0625}{365}\right)^{365 \times 5} \]<br /><br />First, calculate the daily interest rate:<br /><br />\[ \frac{0.0625}{365} \approx 0.00017123 \]<br /><br />Next, add 1 to the daily interest rate:<br /><br />\[ 1 + 0.00017123 \approx 1.00017123 \]<br /><br />Then, raise this to the power of \( 365 \times 5 \):<br /><br />\[ \left(1.00017123\right)^{1825} \approx 1.349858 \]<br /><br />Finally, multiply this by the principal amount:<br /><br />\[ A = 12000 \times 1.349858 \approx 16198.296 \]<br /><br />So, the amount in the account after 5 years will be approximately \( \$16,198.30 \).