Arithmetic growth versus exponential decay I have a kilogram of an element that has a long half-life - say, 1 year - and I put it in a container. Now every day after that I add another kilogram of the element to the container. Does the exponential decay eventually "dominate" or does the amount of the substance in the container increase without bound? I know this should be a simple answer but it's been too long since college...

Looking at this with no equations, a simple thought experiment. If, in fact, it grew without bounds, at some point, you have over 800kg. And with a 1 yr half-life, it will drop to 400kg a year later. Now, during that year, you've only added 365kg, and it's been there an average of 6 months so .707 remains (the square root of 'half'), say 30% gone or about 250 remaining. We now have just 650kg.

The other answers clearly show precise numbers, but when I read your question, it seemed suited to a simple back of envelope approach to simply prove the negative.