You can't argue. It
is 2.22222222.....
To actually be productive, and for those who don't know the math, basically we have what's called an "infinite series" where each successive term in the series (call it x[n+1]) is defined by taking 10% of the previous term. So if you redeem $10, you get $1 back, which you compute from x[n+1] = 0.1 * x[n].
This 10% (or 0.1) term is called the
ratio. The cool thing about this kind of series (a
geometric series) is that you can compute any arbitrary value in the series. So if I want to know what cash back I'll get after redeeming the leftover 10% of my leftover 10% of my leftover 10%, all I would do is multiply my original value ($10) times the
ratio that many (3) number of times. So after 3 redemptions, I would be left with $10 * 0.1^3 = $0.01. I'm rich!!
If you're with me so far, you can see pretty easily that if you can compute any arbitrary value in the series (so the 100th value is just 10 * 0.1^100), you should also be able to compute the total redeemed value up until that point. There's a pretty cool derivation that shows that basically you end up with your initial amount of cash back ($10) some factor of the ratio. It looks something like this:
View attachment 1384
Let's take it a step further. What happens if we redeem our leftovers 1000 times over? 1 million? 1 billion? How about
an infinite number of times?
Okay. For those of you who are confused by the concept of finding the sum of an infinite series of numbers, don't worry about it. The point is that for a sufficiently large number of consecutive redemptions, the amount you get refunded by the A+ gets smaller and smaller and smaller until it is (effectively zero). That's the '1-r^(n+1)' term in that equation. So basically we can approximate this infinite sum by taking the sum of a few terms, which is what
@essenn so beautifully explained. Thus, the A+ is a 2.2% or 2.22% cash back card, depending on who you talk to.
Now for those of you who are still with me and want to take it a step further, let's actually compute this infinite sum. In the limit as n->infinity, the term in the numerator (1-r^(n+1)) basically becomes 1. Why? Because r (our ratio) is a number smaller than 1 and greater than 0 (it's 10% in our case), and if you multiply such a number by itself enough times, it approaches 0 fairly rapidly. So the numerator disappears and we're left with:
View attachment 1385
1/1-(0.1) = 1/0.9 = 1.1111111111111. Multiply that by 2 (which is our 'initial' balance), and we have our 2.22222222....% cash back A+.
Phew. Hope that made sense. Happy earning!
(Image source:
http://mathworld.wolfram.com/GeometricSeries.html)