I have found a formula for dividing numbers in easy steps
What you are using are so called geometric series: $$\frac1{1+n}=\frac1n\frac1{1-(-1/n)}=\frac1n\sum_{k=0}^{\infty}\left(-\frac1n\right)^k$$ thus $$\frac{x}{1+n}=\frac xn\sum_{k=0}^{\infty}\left(-\frac1n\right)^k=\frac{x}n-\frac{x}{n^2}+\frac{x}{n^3}-\dots.$$
Of course the fact that you found the result by observation is quite impressive.