how much more does $1,000 earn in eight years, compounded daily at 3%, than $1,000 over eight years at 3%…

Explanation:

Step1: Recall compound - interest formula

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

Step2: Calculate amount for daily compounding

Given $P = 1000$, $r=0.03$, $n = 365$, and $t = 8$. $A_1=1000(1 +\frac{0.03}{365})^{365\times8}=1000(1+\frac{0.03}{365})^{2920}$. $1+\frac{0.03}{365}\approx1 + 0.0000821918=1.0000821918$. $A_1\approx1000\times(1.0000821918)^{2920}\approx1000\times1.27122 = 1271.22$.

Step3: Calculate amount for semi - annual compounding

Given $P = 1000$, $r = 0.03$, $n = 2$, and $t = 8$. $A_2=1000(1+\frac{0.03}{2})^{2\times8}=1000(1 + 0.015)^{16}$. $(1 + 0.015)^{16}\approx1.26899$. $A_2\approx1000\times1.26899=1268.99$.

Step4: Find the difference

The difference $\Delta A=A_1 - A_2$. $\Delta A=1271.22-1268.99 = 2.23$.

Answer:

$2.23$