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In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. Examples of a geometric sequence are powers ''r''''k'' of a fixed number ''r'', such as 2''k'' and 3''k''. The general form of a geometric sequence is : where ''r'' ≠ 0 is the common ratio and ''a'' is a scale factor, equal to the sequence's start value. ==Elementary properties== The ''n''-th term of a geometric sequence with initial value ''a'' and common ratio ''r'' is given by : Such a geometric sequence also follows the recursive relation : for every integer Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio. The common ratio of a geometric sequence may be negative, resulting in an alternating sequence, with numbers switching from positive to negative and back. For instance :1, −3, 9, −27, 81, −243, ... is a geometric sequence with common ratio −3. The behaviour of a geometric sequence depends on the value of the common ratio. If the common ratio is: * Positive, the terms will all be the same sign as the initial term. * Negative, the terms will alternate between positive and negative. * Greater than 1, there will be exponential growth towards positive or negative infinity (depending on the sign of the initial term). * 1, the progression is a constant sequence. * Between −1 and 1 but not zero, there will be exponential decay towards zero. * −1, the progression is an alternating sequence * Less than −1, for the absolute values there is exponential growth towards (unsigned) infinity, due to the alternating sign. Geometric sequences (with common ratio not equal to −1, 1 or 0) show exponential growth or exponential decay, as opposed to the linear growth (or decline) of an arithmetic progression such as 4, 15, 26, 37, 48, … (with common ''difference'' 11). This result was taken by T.R. Malthus as the mathematical foundation of his ''Principle of Population''. Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression. An interesting result of the definition of a geometric progression is that for any value of the common ratio, any three consecutive terms ''a'', ''b'' and ''c'' will satisfy the following equation: :: where ''b'' is considered to be the ''geometric mean'' between ''a'' and ''c''. ==Geometric series== Computation of the sum 2 + 10 + 50 + 250. The sequence is multiplied term by term by 5, and then subtracted from the original sequence. Two terms remain: the first term, ''a'', and the term one beyond the last, or ''ar''''m''. The desired result, 312, is found by subtracting these two terms and dividing by 1 − 5. A geometric series is the sum of the numbers in a geometric progression. For example: : Letting ''a'' be the first term (here 2), ''m'' be the number of terms (here 4), and ''r'' be the constant that each term is multiplied by to get the next term (here 5), the sum is given by: : In the example above, this gives: : The formula works for any real numbers ''a'' and ''r'' (except ''r'' = 1, which results in a division by zero). For example: : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Geometric progression」の詳細全文を読む スポンサード リンク
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