6.6.56 Combinatorics

Condition: Prove the method of mathematical induction. \ [\ Begin {Array} {l} \ frac {1} {2}-\ frac {2} {2^{2}}+\ frac {3} {2^{3}}-\ frac {4} {4}}}+\ cdots+(-1)^{n+1} {n+1} \ frac {n} {2^{n}} = \\ = \ frac {1} {9} \ left (2+(-1)^{n-1} \ frac {3 n+2} {2^{n} \ right). \ end {Array} \]