The Marshal Conundrum I've thought of a problem which I would like to know the solution to without having to brute force the answer.
The Marshal Conundrum
About
I thought of The Marshal Conundrum after getting the Marshal badge of SFF.SE. I wondered how long it would take me to get 100 comment flags per day on any site just by flagging comments.
The Rules


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*Every flag I submit per day is approved, none are ever declined.

*I'm not earning rep so cant gain flags from rep.

*I gain a flag for every 10th helpful flag.

*Once I've used all my flags for that day I have to wait for the next day


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*Unless I've gained a flag, in which case I can use that flag that day.



Example.
If I wanted to reach 15 flags, it would take me 4 days. This is because:


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*On day one I would use 10 flags. I would then gain a flag and use an 11th.

*On day two I use 11 flags. However my 1 extra from the day before and 9 of my 11 today gave me a new flag. I therefore end the day having submitted 12 flags.

*On day three I use my 12 flags. 3 extra from the day before + 7 from today gain me a flag. I end the day with 13 flags submitted.

*On day four I use my 13 flags. 6 extra from the day before + 4 from today gain me a flag. I use the remaining 10 which earns me another extra flag. I end the day having done 15 flags. 



How many days does it take me to be allowed to cast 100 flags in a day? Is there a rule to generalise it for n flags gained for every m flags cast?
 A: Here is an estimation (expanded from the comments).
Say on a given day we start with $F$ flags. Using them all will reward us with $0.1F$ new flags. Using them again reward us with $0.01F$ flags, and so on. In total we're rewarded $0.111\ldots F=\frac19F$ flags. The next morning our $F$ flags are replenished and we're ready to use $\frac{10}9F$ flags.
So for each day, the number of available flags increase with a factor of $\frac{10}9$, so after $n$ days we have $10\left(\frac{10}9\right)^n$ flags. To find when we reach $100$ flags we solve $10\cdot\left(\frac{10}9\right)^n=100$, which gives $n=21.85$ days. Thinking about what the decimals mean (solving, for instance, $10\cdot\left(\frac{10}9\right)^n=11$ and comparing the answer to the known exact solution) this means that our estimation is that we reach $100$ flags on the $22$nd day of flagging.
This will overestimate our stockpile of flags, because the estimation allows us to use fractional flags that we don't really have and thus gain more rewards than we should. That means it underestimates the number of days it takes to reach a certain size.
