A giant cookie sits on a plate. Cookie Monster eats half the cookie with one bite. With another bite he eats half of the remaining cookie.

Cookie Monster keeps taking bites, eating half the remaining cookie with each bite.

(a) After 5 bites, what fraction of the cookie has Cookie Monster eaten?

(b) How many bites does it take before less than one one-thousandth of the cookie remains?

Answer

After one bite, half the cookie remains. After two bites, one-fourth of the cookie remains. With each bite, the remaining portion of cookie gets halved.
Let's put this information in a table:

After the *n*^{th} bite, the fraction of cookie remaining is .

(a) After 5 bites, the fraction of cookie remaining is

This means Cookie Monster has eaten of the cookie.

(b) We want to find the first value of *n* for which

Since , we need to solve the inequality

Multiplying both sides by 2^{n} and by 1000, we have

1000 < 2^{n}

log _{2} 1000 < *n*

9.966 < *n*

In order for *n* to be a whole number greater than 9.966, we must have *n* = 10. To make sure, we can check the values of *a*_{9} and *a*_{10}.

It takes 10 bites before less than one one-thousandth of the cookie remains.