simplification of powers.

Question

I have the following fraction that I am trying to simplify. $$ \frac{1}{1-(1-x)^i} \bigg[\frac{(1-(1-x)^{i+1}) -(1-(1-x)^{2i})}{1-(1-x)}\bigg] $$ I begin by pulling the denominator out of the fraction and multiplying it with the external factor, cancelling the numerator's 1s. $$ \frac{1}{x-(1-x)^{i+1}} \bigg[(-(1-x)^{i+1})-(1-x)^{2i})\bigg] $$ I then pull $(1-x)^i$ out of the top. $$ \frac{-(1-x)^{i}}{x-(1-x)^{i+1}} \bigg[(1-x)+(1-x)^{i})\bigg] $$ I am not sure how to proceed beyond this point. I believe the final answer should be: $$ \frac{1 - (1-x)^{i+1}}{2-x} $$

Answer 1

We can simplify it as: $$\frac{1}{1-(1-x)^i}[\frac{(1-x)^{2i}-(1-x)^{i+1}}{x}]$$ $$=\frac{(1-x)^i}{1-(1-x)^i}[\frac{(1-x)^i-(1-x)}{x}]$$

There is a mistake in your second step where $x[1-(1-x)^{i}]$ should be $x-x(1-x)^{i}$ and not $x-(1-x)^{i+1}$.