View source for Geometric progression

A sequence of numbers each one of which is equal to the preceding one multiplied by a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442901.png" /> (the denominator of the progression). A geometric progression is called increasing if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442902.png" />, and decreasing if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442903.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442904.png" />, one has a sign-alternating progression. Any term of a geometric progression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442905.png" /> can be expressed by its first term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442906.png" /> and the denominator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442907.png" /> by the formula

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The sum of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g0442909.png" /> terms of a geometric progression (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429010.png" />) is given by the formula

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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429012.png" />, the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429013.png" /> tends to the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429014.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429015.png" /> increases without limit. This number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429016.png" /> is known as the sum of the infinitely-decreasing geometric progression.

The expression

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is the simplest example of a convergent series — a geometric series; the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429018.png" /> is the sum of the geometric series.

The term  "geometric progression"  is connected with the following property of any term of a geometric progression with positive terms: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044290/g04429019.png" />, i.e. any term is the [[Geometric mean|geometric mean]] of the term which precedes it and the term which follows it.