Calculating my improvement via daily compounding

Question

If I improve at something 1% daily what will be my overall improvement after 1 year?

What is the formula to calculate this?

Answer 1

If your current "stock of talent" is $k_{0}$, your stock of talent tomorrow will be $k_{1}=k_{0}(1+0.01)$. Starting from tomorrow's stock $k_{1}$, your stock of talent after an additional day of improvement will be $k_{2}=k_{1}(1+0.01)$. But this is just $k_{2}=k_{0}(1+0.01)(1+0.01)=k_{0}(1+0.01)^2.$ More generally, your stock of talent after $t$ periods of improvement will be $k_{t}=k_{0}(1+0.01)^t$.