\begin{tabular}{|l} Question 3 \ Which compound interest problem yields the most money? \ \hline Compounded quarterly \ O compounded continuously \ Compounded monthly \ \hline Compounded Annually \end{tabular}

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Let the total accumulated amount be $$ A $$, the principal be $$ P $$, the annual interest rate be $$ r $$, and the time in years be $$ t $$.

For interest compounded $$ n $$ times per year, the formula is:

$$
A = P\left(1+\frac{r}{n}\right)^{nt}
$$

For continuously compounded interest, the formula is:

$$
A = P e^{rt}
$$

**Step 1: Recognize the effect of $$ n $$ in the periodic compounding formula**

- When interest is compounded annually, $$ n = 1 $$.
- When compounded quarterly, $$ n = 4 $$.
- When compounded monthly, $$ n = 12 $$.

As $$ n $$ increases, the expression $$\left(1+\frac{r}{n}\right)^{nt}$$ approaches $$ e^{rt} $$.

**Step 2: Comparison of methods**

- For any positive $$ r $$ and $$ t $$, it holds that:
  
  $$
  e^{rt} > \left(1+\frac{r}{n}\right)^{nt}
  $$
  
  for finite values of $$ n $$. Thus, continuous compounding gives a higher final amount than compounding at any finite frequency.

**Step 3: Conclusion**

The compound interest problem that yields the most money is the one with **compounded continuously**.
Continuous compounding yields the most money.

Question 3

Which compound interest problem yields the most money?

Compounded quarterly

O compounded continuously

Compounded monthly

Compounded Annually

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Question 3 Which compound interest problem yields the most money? Compounded quarterly O compounded continuously Compounded monthly Compounded Annually