Formula for consecutively subtracting a percentage n times

Question

I'm looking for a formula to subtract a percentage (e.g. $10\%$) from a number (e.g. $100$) a certain number of times (e.g. $4$) in which case:

$x=100$

$p=0,1$ (i.e. $10\%$)

$n=4$

Doing it manually goes like this:

1. $y_1 = 100 - 10\% = 90$

2. $y_2 = 90 - 10\% = 81$

3. $y_3 = 81 - 10\% = 72.9$

4. $y_4 = 72.9 - 10\% = 65.61$

The formulas for these steps I've managed to figure out are:

1. $y_1 = x - 1\cdot px$
2. $y_2 = x - 2\cdot px + 1\cdot p^2x$
3. $y_3 = x - 3\cdot px + 3\cdot p^2x - 1\cdot 0.1^3x$
4. $y_4 = x - 4\cdot px + 6\cdot p^2x - 4\cdot0.1^3x + 0.1^4x$

The formula 'grows' with each iteration.

Is there a single universal formula that would use $x$, $p$ to find $y_n$ for any $n$?

I suppose in function notation the question would look like $y = f(x,p,n)$.

Answer 1

Note that instead of subtracting $0.1\cdot y_{n-1}$ from $y_{n-1}$, you can simple multiply $(1-0.1)\cdot y_{n-1}$. The factor $(1-0.1)$ doesn't depend on $n$, so the formula becomes much easier: since we multiply by $(1-0.1)$ at each iteration, we effectively multiply by $(1-0.1)^n$ after $n$ iterations.

The formula is

$$y_n=(1-p)^n x.$$