Exercises

Use the principle of mathematical induction in order to prove the following statements.

1.: $\begin{displaymath} \forall \;\; n \in {\rm N}\;\;\;\; 1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{n(n+1)(n+2)}{6} \end{displaymath}$
2.: ( Binomial theorem )

$\begin{displaymath} (a + b)^n = \sum_{m=0}^n \left( \begin{array}{c} n\ m \end{array} \right) \cdot a^{n-m}\cdot b^m, \end{displaymath}$

where

$\begin{displaymath} \left( \begin{array}{c} n\ m \end{array} \right) \end{displaymath}$

denotes

$\begin{displaymath} \frac{n!}{m!(n-m)!}, \end{displaymath}$

with

$\begin{displaymath} r!=1\cdot 2 \cdot 3 \cdot \dots \cdot r\;\;\mbox{ and } 0!=1 \end{displaymath}$
3.: The set of elements has subsets.

Sergey Nikitin 2005-08-28