Exercice

Démontrer par récurrence les propositions suivantes
a) \(\forall n \in \) \( \mathbb{N^*}\), \({2^n} \succ n\)
b) \(\sum\limits_{k = 1}^n k = \) \(\frac{{n\left( {n + 1} \right)}}{2}\) ;
c) \(\sum\limits_{k = 1}^n {{k^2}} = \) \(\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}\) ;
d) \(\sum\limits_{k = 1}^n {{k^3}} = \) \(\frac{{{n^2}{{\left( {n + 1} \right)}^2}}}{4}\) ;
e) \(\sum\limits_{k = 1}^n {\frac{{2k - 1}}{{{2^k}}}} \) \( = 3 - \) \(\frac{{3 + 2n}}{{{2^n}}}\)
f) \(\sum\limits_{k = 1}^n {k\left( {k + 1} \right)} \) \( = \) \(\frac{{n\left( {n + 1} \right)\left( {n + 2} \right)}}{3}\)
g) \(\sum\limits_{k = 1}^n {k{2^{k - 1}}} \) \( = \left( {n - 1} \right){2^n}\) \( + 1\)