Battery charge cycle math

According to Apple, an iPhone battery can go through 400 charge cycles before the capacity diminishes to 80%, assuming nothing happens that is known to degrade the battery's life (i.e, the temperature and pressure are not too high.) A charge cycle is defined as any time when the amount of energy transferred to the battery equals the energy capacity of the battery. So, if you let an iPhone's battery that is 50% full charge all the way to 100%, then do the same thing again when the battery reaches 50%, that counts as a charge cycle.

Assume a typical user charges their iPhone to 100%, then disconnects it from the charger. They then plug it back in when the battery reaches x%. I want to set up a function I can use to calculate how many times the user can plug in the iPhone before the battery capacity diminishes to 80%.

Here's what I've come up with:

slope of the line = 400 / 0.5 = 800
y-intercept = 400

so f(x) = 800x + 400.

Each time the user plugs in their iPhone, they used (100-x)% = 1-(x/100) of a charge cycle. Doing this N times charges the battery N*(1-(x/100)) cycles. When the capacity has been diminished to 80%, this will equal 400. So,

N = 400/(1 - (x/100))

is the formula you're looking for. In the extreme case when x=0, the user completely discharges the battery each time, so N=400/(1-0) = 400, which makes sense. When x=100, the user never charges, so you essentially get N=400/0 = infinity.