In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero 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 rk of a fixed non-zero number r, such as 2k and 3k. The general form of a geometric sequence is where r ≠ 0 is the common ratio and a ≠ 0 is a scale factor, equal to the sequence's start value. The sum of a geometric progression's terms is called a geometric series. The n-th term of a geometric sequence with initial value a = a1 and common ratio r is given by and in general 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 alternating between positive and negative. 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 (→ 0). −1, the absolute value of each term in the sequence is constant and terms alternate in sign. less than −1, for the absolute values there is exponential growth towards (unsigned) infinity, due to the alternating sign.