Question

If each zombie creates 3 new zombies every week what is the expression for how many zombies there will be after `n` weeks?

Answer 1

`z_n=4^n` where `n` is the number of weeks since initial zombie occurrence.

Explanation:

This is basically a compound interest type of formula.

If there are `z_n` zombies in week `n`
then
in week `n+1` we would have
`color(white)("XXX")z_n` zombies from week `n`
`color(white)("XXX")`plus
`color(white)("XXX")3z_n` "fresh" zombies (3 for each zombie from week `n`)
for a total of `4z_n` zombies in week `n+1`

Assuming there was `1` initial zombie (in week `0`) and that no zombies were destroyed,
the zombie population increases by a factor of `4` each week.

`z_0=1`
`z_1=1xx4`
`z_2=1xx4xx4`
`z_3=1xx4xx4xx4`
`z_4=1xx4xx4xx4xx4`
...and so on

with the obvious result that
`color(white)("XXX")z_n=4^n`

Note that since we don't know within any 7 day period when the generation of "new" zombies will occur, we can not reduce this to a function based on days.