The bollworm counter

A model says that a fraction of the living bollworm larvae dies each day on Bt cotton. You will count the survivors, find how long it takes to bring the larvae below a limit, and look up the gene for a pest.

The idea. If a fraction $p$ of the larvae die each day, a fraction $1 - p$ live. After $d$ days, $\text{larvae} \times (1 - p)^d$ remain.

Your tasks.

1. Finish survivors. It gives the number of larvae that live after days days.
2. Finish days_below. It gives the first day on which the number falls below limit.
3. Finish gene_for. It gives the cry gene for the cotton bollworm, or "none" for any other pest.