Recognize geometric sequences

A geometric sequence is understood as a sequence in which each term is formed by multiplying the preceding term by a fixed common ratio, identifiable from listed terms, tables, recursive rules, or forms such as $a_n=ar^{n-1}$ . The pattern may increase, decrease, remain constant, or alternate when the ratio is fractional, one, or negative; learners distinguish this multiplicative structure from the constant differences of arithmetic sequences, without requiring infinite-series convergence or more advanced generalizations.

Detailed Explanation: Recognize geometric sequences

A geometric sequence is a sequence in which each term is found by multiplying the previous term by the same number. This fixed number is called the common ratio, written as $r$ .

To recognize one:

Divide each term by the term immediately before it.
Check whether the quotient is always the same.
If it is, the sequence is geometric.

Example

Determine whether the sequence

81,\ 27,\ 9,\ 3,\ldots

is geometric.

Divide each term by the preceding term:

\frac{27}{81}=\frac{1}{3}

\frac{9}{27}=\frac{1}{3}

\frac{3}{9}=\frac{1}{3}

The ratio is always $\frac{1}{3}$ , so the sequence is geometric. Its common ratio is

r=\frac{1}{3}.

This means each term is found by multiplying the previous term by $\frac{1}{3}$ :

81\cdot\frac{1}{3}=27,\qquad 27\cdot\frac{1}{3}=9.

Do not confuse this with an arithmetic sequence, where the same number is added or subtracted each time. Here, the differences are not constant, but the multiplying factor is constant.

Learn by doing: Recognize geometric sequences

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Patterning - Rule from Equation for Geometric Pattern

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