find the interest earned on $25,000 invested for 6 years at 7% interest compounded as follows.\na…

Explanation:

Step1: Recall compound - interest formula

The compound - interest formula is $A = P(1+\frac{r}{n})^{nt}$ for non - continuous compounding and $A = Pe^{rt}$ for continuous compounding, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal form), $n$ is the number of times compounded per year, $t$ is the number of years, and $A$ is the final amount. The interest earned $I=A - P$. Given $P = 25000$, $r=0.07$, and $t = 6$.

Step2: Calculate for annual compounding ($n = 1$)

$A=P(1 +\frac{r}{n})^{nt}=25000(1+\frac{0.07}{1})^{1\times6}=25000(1.07)^{6}$. $A = 25000\times1.500730351\approx37518.26$. $I=A - P=37518.26−25000 = 12518.26$.

Step3: Calculate for semi - annual compounding ($n = 2$)

$A=P(1+\frac{r}{n})^{nt}=25000(1+\frac{0.07}{2})^{2\times6}=25000(1.035)^{12}$. $A = 25000\times1.511601943\approx37790.05$. $I=A - P=37790.05−25000 = 12790.05$.

Step4: Calculate for quarterly compounding ($n = 4$)

$A=P(1+\frac{r}{n})^{nt}=25000(1+\frac{0.07}{4})^{4\times6}=25000(1.0175)^{24}$. $A = 25000\times1.517856188\approx37946.40$. $I=A - P=37946.40−25000 = 12946.40$.

Step5: Calculate for monthly compounding ($n = 12$)

$A=P(1+\frac{r}{n})^{nt}=25000(1+\frac{0.07}{12})^{12\times6}=25000(1+\frac{0.07}{12})^{72}$. $A = 25000\times1.521997217\approx38049.93$. $I=A - P=38049.93−25000 = 13049.93$.

Step6: Calculate for continuous compounding

$A = Pe^{rt}=25000e^{0.07\times6}=25000e^{0.42}$. $A = 25000\times1.521962215\approx38049.06$. $I=A - P=38049.06−25000 = 13049.06$.

Answer:

a. $$12518.26$ b. $$12790.05$ c. $$12946.40$ d. $$13049.93$ e. $$13049.06$