6.6.56 Combinatorics

condition: Prove by mathematical induction. \[ \begin{array}{l} \frac{1}{2}-\frac{2}{2^{2}}+\frac{3}{2^{3}}-\frac{4}{2^{4}}+\cdots+(-1)^{n+1} \frac{n}{2^{n}}= \\ =\frac{1}{9}\left(2+(-1)^{n-1} \frac{3 n+2}{2^{n}}\right) . \end{array}\]