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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.
The geometric series is an infinite series derived from a special type of sequence called a geometric progression.This means that it is the sum of infinitely many terms of geometric progression: starting from the initial term , and the next one being the initial term multiplied by a constant number known as the common ratio .
It is useful to figure out which summation methods produce the geometric series formula for which common ratios. One application for this information is the so-called Borel-Okada principle: If a regular summation method assigns = to / for all in a subset of the complex plane, given certain restrictions on , then the method also gives the analytic continuation of any other function () = = on ...
More generally, for any initial value in the real numbers and a real number common ratio between -1 and 1, a geometric progression () converges linearly with rate | | and the sequence of partial sums of a geometric series (=) also converges linearly with rate | |.
Today, a more standard phrasing of Archimedes' proposition is that the partial sums of the series 1 + 1 / 4 + 1 / 16 + ⋯ are: + + + + = +. This form can be proved by multiplying both sides by 1 − 1 / 4 and observing that all but the first and the last of the terms on the left-hand side of the equation cancel in pairs.
The Kepler triangle is a right triangle whose sides are in geometric progression. If the sides are formed from the geometric progression a, ar, ar 2 then its common ratio r is given by r = √ φ where φ is the golden ratio. Its sides are therefore in the ratio 1 : √ φ : φ. Thus, the shape of the Kepler triangle is uniquely determined (up ...
A geometric series [20] [21] is one where each successive term is produced by multiplying the previous term by a constant number (called the common ratio in this context). For example: 1 + 1 2 + 1 4 + 1 8 + 1 16 + ⋯ = ∑ n = 0 ∞ 1 2 n = 2. {\displaystyle 1+{1 \over 2}+{1 \over 4}+{1 \over 8}+{1 \over 16}+\cdots =\sum _{n=0}^{\infty }{1 ...
Exceptionally, the golden ratio is equal to the limit of the ratios of successive terms in the Fibonacci sequence and sequence of Lucas numbers: [42] + = + =. In other words, if a Fibonacci and Lucas number is divided by its immediate predecessor in the sequence, the quotient approximates φ {\displaystyle \varphi } .