Function for "active users" over time, given a constant "new users per day", and an exponential decay of new users who keep the app after "x" days Forgive me if this is basic, it has been a while since I have used maths like this.
Given:
$install\ rate= 1\ new\ user\ per\ day$
$retention\ rate = 0.85^x$
The "retention rate" is percentage of new users who keep the app after "x" days. In my example, about 30% of the users keep the app after 7 days:
$0.85^7 = 0.32$
and basically no one keeps using it after 30 days. It decays towards zero because this app isn't used long term.
How would I create a function for active users over time, and will this function approach a positive finite limit? How would I calculate this steady-state value?
And a follow up question: My "retention rate" model isn't very good, I tried to fit my data to an exponential equation, but I only had one parameter to play with and $0.85^x$ is the best I could do. The actual data looks like exponential decay going towards zero, but 30% of users keeps the app after 1 day, 10% after 7 days, 3% after 30 days, and close to 0% after 90 days. I'm guessing I need a polynomial model to fit this data with, but I do not know how to do this. What is an equation that fits this data, and how does the answer to the first question change when using this new model?
 A: You're asking about a sequence given by $a_0 = 0$ and $a_n = 0.85 a_{n-1} + 1$. The easiest way to understand what will happen in the limit is to say that if you were already at the limiting value $a_n = A$ then you would stay there, i.e. $a_{n+1} = A$. In other words, you'd need $A = a_{n+1} = 0.85 a_n + 1 = 0.85 A + 1$. So the limiting value must satisfy $A = 0.85A + 1$ which is easily solved: $A = \frac 1 {0.15} \approx 6.67$.
Alternatively, you could find a formula for the exact value of $a_n$. The system $a_n = 0.85 a_{n-1} + 1$ and $a_0 = 0$ is called a "non-homogeneous linear recurrence relation with constant coefficients". There are standard methods for finding exactly solutions, e.g. see Wikipedia. Alternatively, you can just ask WolframAlpha for the solution like this. Either way, I get
$$a_n = \frac{20}{3} \left(1 - \left(\frac{17}{20}\right)^n \right).$$
If you tried the same thing for other values of $a_0$, you'd get
$$a_n = \frac{20}{3} \left(1 - \left(\frac{17}{20}\right)^n \right) + \left(\frac{17}{20}\right)^n a_0.$$
As $n \to \infty$, most of the terms go to 0 in those formulas, and you're let with $\frac{20}{3}$ which agrees with the result above. Regardless of which method we pick, we see that $a_0$, the number of users you started with, has no effect on the limit.

Regarding the followup: I actually think that with just a few data points, exponential decay with 1 parameter will be about as good as you can do. Trying to fit a multi-parameter model to 3 data points would likely just create weird overfitting effects. One thing I might try is to adjust the decay parameter $0.85$ so that the limit comes out about right. For example, if the real-life longterm average is around 10, then the correct decay constant would be 0.9 (you could check this using one of the methods above).
On the other hand, if you have access to some more data you could try fitting a slightly more complicated model. If you have more install/uninstall data available, you could try fitting a model where the probability of a user uninstalling depends on how long they've had the app so far. For example, you could start from the assumption that the uninstall probability is $70\%$ for days 1-2, but then users who still have the app on day 3 will settle into a slower decay with some unknown rate. You could solve for that rate by the same technique again - making the model's limit prediction fit with the real-life data.
That was just an example. You could also try coming up with some different assumption for how the decay changes over time and then see where that leads you. If you do end up building multiple models, you should make sure to compare them carefully at the end to decide which is the most useful. Remember to be wary of overfitting effects if one model gives a better fit but also has more parameters than the other.
