Geometric sequences.

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

Example 3.4.1   Natural radioactivity determines geometric sequences whose quotient is equal to $ \frac 12$ .

Explicit formula:

\bgroup\color{blue}$ \forall n \in \mathbb{N}, \; u_n=u_0 \cdot q^n$\egroup .

Proposition 3.4.2 (Sum of the first successive $ n$ terms)       

If \bgroup\color{blue}$ q \neq 1$\egroup , \bgroup\color{blue}$ \underset{k=0}{\overset{n}{\sum}} u_k = u_0 \frac {1-q^{n+1}}{1-q}$\egroup



Noah Dana-Picard 2007-12-28