Explanation:
To determine the single deposit Lisa would need to make now to have $50,000 in 10 years with an account that pays 4.1% interest compounded daily, you can use the formula for compound interest:
\[A = P(1 + \frac{r}{n})^{nt}\]
Where:
- A is the future amount ($50,000 in this case).
- P is the principal amount (the initial deposit we want to find).
- r is the annual interest rate (4.1% or 0.041 as a decimal).
- n is the number of times the interest is compounded per year (daily, so 365 times a year).
- t is the number of years (10 in this case).
Now, we can solve for P:
\[50,000 = P(1 + \frac{0.041}{365})^{365 * 10}\]
First, calculate the values inside the parentheses:
\[1 + \frac{0.041}{365} \approx 1.00011232876\]
Next, calculate the exponent:
\[365 * 10 = 3650\]
Now, the equation becomes:
\[50,000 = P(1.00011232876)^{3650}\]
To isolate P, divide both sides of the equation by \((1.00011232876)^{3650}\):
\[P = \frac{50,000}{(1.00011232876)^{3650}}\]
Calculating this gives you the initial deposit Lisa would need to make now.