Where am I getting..................2(3/2)(5/4)(7/6)(11/10)?...............ten and 11......................geometric sequence..............a scaling down of 2 for successive powers of 10, Gauss saw a pattern.........2 and 11, beg and end of the pf.........11/10..............a geo progression.........which is where Euler got his idea to equate the two........one series based off of addition........the other off of multiplication..........11 + 10 = 21...............1/2 + 21i........the location of the 2nd zero......................

Geometric progression

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Diagram illustrating three basic geometric sequences of the pattern 1(rⁿ⁻¹) up to 6 iterations deep. The first block is a unit block and the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively.

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

${\displaystyle a,\ ar,\ ar^{2},\ ar^{3},\ ar^{4},\ \ldots }$

where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence's start value.

Contents

 [hide] 

• 1 Elementary properties
• 2 Geometric series
  • 2.1 Derivation
  • 2.2 Related formulas
  • 2.3 Infinite geometric series
  • 2.4 Complex numbers
• 3 Product
• 4 Relationship to geometry and Euclid's work
• 5 See also
• 6 References
• 7 External links

Elementary properties[edit]

The n-th term of a geometric sequence with initial value a and common ratio r is given by

${\displaystyle a_{n}=a\,r^{n-1}.}$

Such a geometric sequence also follows the recursive relation

${\displaystyle a_{n}=r\,a_{n-1}}$ for every integer ${\displaystyle n\geq 1.}$

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: