Arithmetic
\(2, 5, 8, 11, \ldots \rightarrow d = 3, a_{10} = 29\)
Folgen & Reihen-Rechner
Berechnen Sie arithmetische und geometrische Folgen, finden Sie n-te Terme, Summen von Reihen und analysieren Sie Konvergenz
Ergebnisse
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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