# Why is this way of calculating mean life of radioactive atoms incorrect?

Your mistake is here:

Which means, 1/2 of the atoms will have a life of t

Half of the the remaining half or 1/4 of the atoms will have a life of 2t and so on.

The corrected statement is:

Which means, 1/2 of the atoms will have a life $$\le t$$

Half of the the remaining half or 1/4 of the atoms will have a life between $$t$$ and $$2t$$ and so on.

Which means, $$1/2$$ of the atoms will have a life of $$t$$

is not a correct statement as some of them will have a lifetime of almost zero, some $$\frac 12t$$ some $$\frac{199}{200}t$$ etc, so you have overestimated the time that the atoms live.

To see how it should be done correctly read the answer to Mean life of radioactive substance.