Consider a substance which disintegrates at a rate proportional to the amount present. Let denote the amount of the substance present at time . Define the th life of the substance for a positive integer as the number such that
- Prove that the th life is the same for any given sample of the substance (i.e., prove that the th life is invariant under the starting size of the sample), and compute in terms of and the decay constant .
- For given real numbers and , prove we can write the function as
and determine the function .
- Proof. Since
This is independent of the initial sample (since it depends only on the th life and the decay constant ). Further, computing the th life in terms of and the decay constant we have
- Proof. Let