the same amount of principal is invested in different accounts earning the same interest rate. which of the…

Explanation:

Step1: Recall interest - earning concepts

Simple interest formula is $A = P(1+rt)$ where $P$ is principal, $r$ is annual interest rate and $t = 1$ year. Compounded - annually formula is $A=P(1 + r)^1=P(1 + r)$. Compounded - daily formula is $A = P(1+\frac{r}{n})^{nt}$, with $n = 365$ (number of days in a year) and $t = 1$, so $A=P(1+\frac{r}{365})^{365}$. Account with no interest has $A = P$.

Step2: Compare the formulas

Let's assume $P>0$ and $r>0$. For simple interest $A_{s}=P(1 + r)=P+Pr$. For annual - compounding $A_{a}=P(1 + r)=P+Pr$. For daily - compounding $A_{d}=P(1+\frac{r}{365})^{365}$. By the formula of compound interest, the more frequently interest is compounded, the higher the accumulated amount. Since $(1+\frac{r}{365})^{365}>1 + r$ for $r>0$ (using the fact that the function $f(n)=(1+\frac{x}{n})^{n}$ is an increasing function of $n$ for $x>0$ and $n>0$ and its limit as $n\rightarrow\infty$ is $e^{x}$).

Answer:

D. An account earning interest compounded daily