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Geometric sequences

Geometric sequences

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In a geometric sequence, all terms are multiplied or divided by the same value. For example, the sequence 2, 4, 8, 16, ... is geometric, as each term is twice the previous term.

Basics of geometric sequences

The term to term rule in a geometric sequence is multiplying or dividing by the same value, also known as the common ratio (r). The common ratio is determined by dividing one term by the previous term.

Terms in a geometric sequence

The nth term, or general term, of a geometric sequence is given by a formula (see image).

nth term of geometric sequences

A geometric sequence (i.e. a list of terms) can also be represented as a geometric series (a sum of terms). Sigma notation (see image) is used to calculate the sum of a finite number of terms of a geometric sequence.

Geometric series

An infinite series has an infinite number of terms. The partial sum (Sβ‚™) of these terms is the sum of the first n terms. If Sβ‚™ approaches a finite limit (known as the sum to infinity), the series is convergent. In a geometric series, if |r|< 1, the sum to infinity can be calculated with a formula (see image).

Convergent sequences

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