Once a week Katherine Baker makes her famous lemon cupcakes

Once a week. Katherine Baker makes her famous lemon cupcakes. The first week that she makes the recipe, she uses the full 2 cups of sugar the recipe calls for. Each week after that, she reduces foe amount of sugar by one fifth. If Katherine becomes immortal and bakes cupcakes every week for all eternity, how much total sugar will she use?

Solution

For first week, Catherine used 2 cups of Sugar

Second week, Catherine will use 2 * 4/5 = 8/5 cups of sugar, since she is reducing the amount by 1/5th

Hence the amount of sugar used by Catherine

=> 2 + 2 * (4/5) + 2 * (4/5)^2 + ....

=> 2(1 + 4/5 + (4/5)^2 + ....)

Using the infinite GP formula

Summation of infinite term with common difference between 0 < r < 1 is equal to a/(1-r)

=> 2 * 1/(1-4/5)

=> 2 * 5

=> 10 cups

Hence catherine will use 10 cups of sugar in his life time