A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 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
a , a r , a r 2 , a r 3 , a r 4 , … {\displaystyle a,\ ar,\ ar^{2},\ ar^{3},\ ar^{4},\ \ldots }
where r is the common ratio and a is the initial value. The sum of a geometric progression's terms is called a geometric series. Because each two successive numbers in the progression have the same proportion, the terms in a geometric series are also said to be in continued proportion, especially in the context of ancient Greek mathematics, where geometric progressions were described in this way rather than through powers of the common ratio.
Properties The nth term of a geometric sequence with initial value a = a1 and common ratio r is given by
a n = a r n − 1 , {\displaystyle a_{n}=a\,r^{n-1},}
and in general
a n = a m r n − m . {\displaystyle a_{n}=a_{m}\,r^{n-m}.}
Geometric sequences satisfy the linear recurrence relation
a n = r a n − 1 {\displaystyle a_{n}=r\,a_{n-1}} for every integer n > 1. {\displaystyle n>1.}
This is a first-order, homogeneous linear recurrence with constant coefficients. Geometric sequences also satisfy the nonlinear recurrence relation
a n = a n − 1 2 / a n − 2 {\displaystyle a_{n}=a_{n-1}^{2}/a_{n-2}}
for every integer n > 2 {\displaystyle n>2} . This is a second-order nonlinear recurrence with constant coefficients. When the common ratio of a geometric sequence is positive, the sequence's terms will all share the sign of the first term. When the common ratio of a geometric sequence is negative, the sequence's terms alternate between positive and negative; this is called an alternating sequence. For instance, the sequence 1, −3, 9, −27, 81, −243, ... is an alternating geometric sequence with an initial value of 1 and a common ratio of −3. When the initial term and common ratio are complex numbers, the terms' complex arguments follow an arithmetic progression. If the absolute value of the common ratio is smaller than 1, the terms will decrease in magnitude and approach zero via an exponential decay. If the absolute value of the common ratio is greater than 1, the terms will increase in magnitude and approach infinity via an exponential growth. If the absolute value of the common ratio equals 1, the terms will stay the same size indefinitely, though their signs or complex arguments may change. Geometric progressions show exponential growth or exponential decline, as opposed to arithmetic progressions showing linear growth or linear decline. This comparison was taken by T.R. Malthus as the mathematical foundation of his An Essay on the Principle of Population. The two kinds of progression are related through the exponential function and the logarithm: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression yields an arithmetic progression. The relation that the logarithm provides between a geometric progression in its argument and an arithmetic progression of values, prompted A. A. de Sarasa to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in prosthaphaeresis, leading to the term "hyperbolic logarithm", a synonym for natural logarithm.
Geometric series
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