In a geometric sequence, the square of the middle term of any three consecutive terms is equal to the product of the other two terms. For example, examine the sequence 2 , − 6 , 18 , − 54 , … . 2 comma negative 6 comma . 18 comma . negative 54 comma dot dot dot .

( − 6 ) 2 = 2 · 18 = 36 _ 2   | , − 6   | , 18   | , − 54   | , … ¯ 18 2 = ( − 6 ) ( − 54 ) = 324 table with 5 rows and 1 column , row1 column 1 , open , negative 6 , close squared . equals 2 middle dot 18 equals 36 , row2 column 1 , modified under bar with

In an arithmetic sequence, recall that the middle term of any three consecutive terms is the arithmetic mean of the other two terms.

The geometric mean of two positive numbers x and y is x y . square root of x y end root , .

Note that the geometric mean is positive by definition. While there are two possible values for the missing term in the geometric sequence 3 , □ , 12 , … , 3 comma white square comma 12 comma dot dot dot comma  there is only one geometric mean. The geometric mean is one possible value to fill in the geometric sequence. The opposite of the geometric mean is the other.

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