How do you solve the geometric series 24-12+6-3+...

Question

This is confusing stuff, why do we need to learn this anyway??

Answer 1

The common multiple between consecutive terms is -1/2. Let the sum of the series be S. If we multiply the series by 1/2 we get another series: 12-6+3-1.5+... If we add this series to the original we get 24 because all the other terms cancel out and since the series runs to infinity, the "last" term tends to zero. We added the series to half of itself, S+(1/2)S=3S/2=24, so S=(2/3)24=16. So the sum to infinity is 16 (believe it or not).