Math 2.0

definition

Geometric sequence

Definition

Let aa and rr be real numbers. The geometric sequence with initial term aa and common ratio rr is the real sequence (a rn)n∈N(a\,r^n)_{n \in \mathbb{N}}. Its terms are a,ar,ar2,ar3,…a, a r, a r^2, a r^3, \dots, and each term is rr times the term before it.

Remarks

Many texts require a≠0a \ne 0 and r≠0r \ne 0, so that the quotient of two consecutive terms is always rr. The sequence 1,2,4,8,…1, 2, 4, 8, \dots has a=1a = 1 and r=2r = 2. The sequence 1,1/2,1/4,…1, 1/2, 1/4, \dots has r=1/2r = 1/2. For ∣r∣<1|r| < 1, the sequence converges to 00. For r>1r > 1 and a>0a > 0, it tends to infinity. The sum of the terms is the geometric series.

Explanations

L1 human

In a geometric sequence, you go from one term to the next by multiplication with the same number rr. If r=3r = 3 and you start at 22, you get 2,6,18,54,…2, 6, 18, 54, \dots.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinition
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Sources

  1. Web https://en.wikipedia.org/wiki/Geometric_progression, lead section

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