## Compounding with a 100% Interest Rate

A man knows that money in an account where interest is compounded semi-annually will earn interest faster than money in an account where interest is compounded annually. He wonders how much interest can be earned by compounding it more and more often. In this problem we investigate his question.

For ease of computation, let's suppose the man invests \$1 at a 100% interest rate. If his interest is compounded annually, his year-end balance will be: $$\begin{align} \$1 + 100 \% \cdot \$1 &= \$1 + 1.00 \cdot \$1\\ & = \$1(1+1) \\ &= \$2. \end{align}$$

If his interest is compounded semi-annually, he earns half the annual interest at mid-year, and so his mid-year balance is: $$ \begin{align} \$1 + \frac{100 \%}{2} \cdot \$1 &= \$1 + \frac{1.00}{2} \cdot \$1 \\ &= \$1 \cdot\left(1+\frac{1}{2}\right) \\ &= \$ 1.5. \end{align}$$ At year-end he earns the other half of his annual interest giving him a year-end balance of:

- Find the man's year-end balance if his interested is compounded quarterly.
- Write an expression which gives the man's year-end balance in terms of the number of times the interest is compounded, $n$.
- Now we'll investigate what happens to the year-end balance as we compound the interest more and more. This means that we want to increase the value of $n$. Complete the table below to help you see what happens to the end of year balance as $n$ becomes larger and larger. Round values to the 5th decimal place.
$n$ Year end account balance when interest is compounded $n$ times 10 100 1000 10,000 100,000 - Based on the results of your table, does it appear that the man can make an unlimited amount of money off of his $1 investment if the bank compounds the interest more and more often? Explain.