Arithmetic sequences.

A sequence $ (u_n)_{n \geq 0}$ is arithmetic if there exists a real number $ d$ , called the difference of the sequence, such that: $ \forall n, \; u_{n+1}=u_n + d$ .

Example 3.3.1   The sequence of all the natural numbers is an arithmetic sequence, whose first term is 0 and whose difference is 1.

Explicit formula:

\bgroup\color{blue}$ \forall n \in \mathbb{N}, \; u_n=u_0+nd$\egroup .

Proposition 3.3.2 (Sum of the first successive $ n$ terms)       

\bgroup\color{blue}$ \underset{k=0}{\overset{n}{\sum}} u_k = \frac 12 [ 2u_0+nd]$\egroup



Noah Dana-Picard 2007-12-28