Once a week Mrs. Baker makes sugar cookies. The first week she makes the recipe, she uses the full 2 cups of sugar called for. Each week after that, she reduces the amount of sugar by one third.

(a) How much sugar does she use for the cookies on the fifth week?

(b) How much sugar does she use for cookies over half a year?

(c) If Mrs. Baker became immortal and baked cookies every week for all eternity, how much sugar would she use?

Answer

If Mrs. Baker reduces the amount of sugar by each week, that means she *keeps* of the previous amount of sugar. So on the second week she uses cups of sugar, and so on. Let's make a table.

(a) From the table, we can see that on the fifth week Mrs. Baker uses

cups of sugar.

(b) The amount of sugar Mrs. Baker uses each week is described by a geometric series with *a* = 2 and . We want to know how much sugar she uses over half a year, or 26 weeks. This means we need to take the partial sum of the geometric series with *n* = 26. We get

She uses almost 6 cups of sugar.

(c) To find the amount of sugar Mrs. Baker uses if she bakes cookies every week for all eternity, we need to find the infinite sum of the geometric series:

Mrs. Baker uses 6 cups of sugar if she bakes cookies every week for all eternity - and from (b) we know that almost all that sugar is used in the first half year of baking!