<s>[INST] Can't prove an equation using induction I have an equation
$$
\frac{1}{2} + \frac{2}{2^2} + \frac{3}{2^3} + \cdots + \frac{n}{2^n} = 2 - \frac{n + 2}{2^n}
$$
Below is what I have already done:
$$
\frac{1}{2} + \frac{2}{2^2} + \frac{3}{2^3} + \cdots + \frac{n}{2^n} + \frac{n + 1}{2^{n+1}} = 2 - \frac{n + 2}{2^n} + \frac{n + 1}{2^{n+1}}=
2 - \frac{3n + 5}{2^{n + 1}} 
$$
I want the right part of the equation to be transformed to $$ 2-\frac{n + 1 + 2}{2^{n+1}}$$
Maybe do I do something wrong? I will be appreciated for any help.
 [/INST] You forgot about the negative sign:
$$
-\frac{n+2}{2^{n}}+\frac{n+1}{2^{n+1}}=\frac{-(2n+4)+n+1}{2^{n+1}}=\frac{-n-3}{2^{n+1}}=-\frac{(n+1)+2}{2^{n+1}}.
$$
</s>