what lump sum must be invested at 12%, compounded monthly, for the investment to grow to $72,000 in 5…

Explanation:

Step1: Identify compound - interest formula

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$, where $A$ is the final amount, $P$ is the principal (initial investment), $r$ is the annual interest rate (in decimal form), $n$ is the number of times interest is compounded per year, and $t$ is the number of years. We want to solve for $P$, so $P=\frac{A}{(1 +\frac{r}{n})^{nt}}$.

Step2: Convert values to appropriate form

The annual interest rate $r = 0.12$ (since $12%=0.12$), the number of times compounded per year $n = 12$ (compounded monthly), the number of years $t = 5$, and the final amount $A=72000$.

Step3: Substitute values into the formula

$P=\frac{72000}{(1+\frac{0.12}{12})^{12\times5}}$. First, calculate the value inside the parentheses: $\frac{0.12}{12}=0.01$, then $1 + 0.01=1.01$. Next, calculate the exponent: $12\times5 = 60$. So, $(1.01)^{60}\approx1.816696698$. Then, $P=\frac{72000}{1.816696698}\approx39620.09$.

Answer:

$39620.09$