Arithmetic
\(2, 5, 8, 11, \ldots \rightarrow d = 3, a_{10} = 29\)
Calculatrice de suites et séries
Calculer les suites arithmétiques et géométriques, trouver les nièmes termes, les sommes de séries et analyser la convergence
Résultats
Entrez les valeurs et cliquez sur Calculer pour voir le résultat.
Theory & Formula
Sequences are ordered lists of numbers. A series is the sum of the terms of a sequence.
Arithmetic Sequences:
- Each term differs by a constant \((d)\)
- nth term: \(a_n = a_1 + (n - 1)d\)
- Sum: \(S_n = \frac{n}{2}[2a_1 + (n - 1)d]\) or \(S_n = \frac{n}{2}(a_1 + a_n)\)
- Linear growth pattern
Geometric Sequences:
- Each term is multiplied by a constant \((r)\)
- nth term: \(a_n = a_1 \times r^{n-1}\)
- Sum: \(S_n = \frac{a_1(1 - r^n)}{1 - r}\) for \(r \neq 1\)
- Converges to \(S_\infty = \frac{a_1}{1 - r}\) when \(|r| < 1\)
- Exponential growth/decay pattern
\(\text{Arithmetic: } a_n = a_1 + (n-1)d \quad | \quad \text{Geometric: } a_n = a_1r^{n-1}\)
Worked Examples
Geometric (Converges)
\(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots \rightarrow r = \frac{1}{2}, S_\infty = 2\)
Geometric (Diverges)
\(1, 2, 4, 8, \ldots \rightarrow r = 2\), diverges