# Geometric sequence meaning and formula

In mathematicsa geometric progressionalso known as a geometric sequenceis 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, Similarly 10, 5, 2. Examples of a geometric sequence are powers r k of a fixed number rsuch as 2 k and 3 k. The general form of a geometric sequence is. Such a geometric sequence also follows the recursive relation.

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. The behaviour of a geometric sequence depends on the value of the common ratio. If the common ratio is:. This result was taken by T. 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 the geometric progression is that any three consecutive terms ab and c will satisfy the following equation:.

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The sequence is multiplied term by term by 5, and then subtracted from the original sequence. Two terms remain: the first term, aand the term one beyond the last, or ar m. A geometric series is the sum of the numbers in a geometric progression. For example:. Letting a be the first term here 2n be the number of terms here 4and r be the constant that each term is multiplied by to get the next term here 5the sum is given by:.In mathematicsa geometric series is a series with a constant ratio between successive terms. For example, the series. Geometric series are among the simplest examples of infinite series with finite sums, although not all of them have this property.

Historically, geometric series played an important role in the early development of calculusand they continue to be central in the study of convergence of series. Geometric series are used throughout mathematics, and they have important applications in physicsengineeringbiologyeconomicscomputer sciencequeueing theoryand finance.

The terms of a geometric series form a geometric progressionmeaning that the ratio of successive terms in the series is constant. This relationship allows for the representation of a geometric series using only two terms, r and a. The term r is the common ratio, and a is the first term of the series.

As an example the geometric series given in the introduction. The sum of a geometric series is finite as long as the absolute value of the ratio is less than 1; as the numbers near zero, they become insignificantly small, allowing a sum to be calculated despite the series containing infinitely many terms.

The sum can be computed using the self-similarity of the series. This new series is the same as the original, except that the first term is missing. A similar technique can be used to evaluate any self-similar expression. One can derive the formula for the sum, sas follows:.

As n goes to infinity, the absolute value of r must be less than one for the series to converge. The sum then becomes. The formula also holds for complex rwith the corresponding restriction, the modulus of r is strictly less than one.

We can prove that the geometric series converges using the sum formula for a geometric progression :. Convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series. Consider the function. For example:. The formula works not only for a single repeating figure, but also for a repeating group of figures.

Note that every series of repeating consecutive decimals can be conveniently simplified with the following:. That is, a repeating decimal with repeat length n is equal to the quotient of the repeating part as an integer and 10 n - 1.

Archimedes used the sum of a geometric series to compute the area enclosed by a parabola and a straight line. His method was to dissect the area into an infinite number of triangles. The first term represents the area of the blue triangle, the second term the areas of the two green triangles, the third term the areas of the four yellow triangles, and so on.

Simplifying the fractions gives. This computation uses the method of exhaustionan early version of integration. Using calculusthe same area could be found by a definite integral.

In the study of fractalsgeometric series often arise as the perimeterareaor volume of a self-similar figure. For example, the area inside the Koch snowflake can be described as the union of infinitely many equilateral triangles see figure.In a Geometric Sequence each term is found by multiplying the previous term by a constant. Each term except the first term is found by multiplying the previous term by 2.

We use "n-1" because ar 0 is for the 1st term. Each term is ar kwhere k starts at 0 and goes up to n It is called Sigma Notation. It says "Sum up n where n goes from 1 to 4. The formula is easy to use And, yes, it is easier to just add them in this exampleas there are only 4 terms.

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But imagine adding 50 terms On the page Binary Digits we give an example of grains of rice on a chess board. The question is asked:.

Which was exactly the result we got on the Binary Digits page thank goodness! Let's see why the formula works, because we get to use an interesting "trick" which is worth knowing.

## Arithmetic & Geometric Sequences

All the terms in the middle neatly cancel out. Which is a neat trick. On another page we asked "Does 0. So there we have it Geometric Sequences and their sums can do all sorts of amazing and powerful things. Hide Ads About Ads. Geometric Sequences and Sums Sequence A Sequence is a set of things usually numbers that are in order. Geometric Sequences In a Geometric Sequence each term is found by multiplying the previous term by a constant.

Example: 1, 2, 4, 8, 16, 32, 64, Example: 10, 30, 90,Example: 4, 2, 1, 0.

Algebra - Sequences And Series (1 of 6) An Introduction

Geometric Sequences are sometimes called Geometric Progressions G. It is called Sigma Notation called Sigma means "sum up" And below and above it are shown the starting and ending values: It says "Sum up n where n goes from 1 to 4. Example: Sum the first 4 terms of 10, 30, 90,The question is asked: When we place rice on a chess board: 1 grain on the first square, 2 grains on the second square, 4 grains on the third and so on,Your browser seems to have Javascript disabled. We're sorry, but in order to log in and use all the features of this website, you will need to enable JavaScript in your browser.

To test whether a sequence is a geometric sequence or not, check if the ratio between any two consecutive terms is constant:.

The geometric mean between two numbers is the value that forms a geometric sequence together with the two numbers. Important: remember to include both the positive and negative square root. The geometric mean generates two possible geometric sequences:. This is a lesson from the tutorial, Sequences and Series and you are encouraged to log in or registerso that you can track your progress.

Register or login to make commenting easier. Save my name, email, and website in this browser for the next time I comment. Toggle navigation. Search Log In. The General Term for a Geometric Sequence. To do 4 min read. Day n No. Download the article for free at Siyavula. Share Thoughts. Example: A Flu Epidemic. Share Thoughts Post Image. Cancel Reply. Add Math. Math Editor. Edit math using TeX:. Math preview:. Close Insert Math.Intro Examples Arith. Series Geo.

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The two simplest sequences to work with are arithmetic and geometric sequences. An arithmetic sequence goes from one term to the next by always adding or subtracting the same value. For instance, 2, 5, 8, 11, 14, The number added or subtracted at each stage of an arithmetic sequence is called the "common difference" dbecause if you subtract that is, if you find the difference of successive terms, you'll always get this common value.

A geometric sequence goes from one term to the next by always multiplying or dividing by the same value. So 1, 2, 4, 8, 16, The number multiplied or divided at each stage of a geometric sequence is called the "common ratio" rbecause if you divide that is, if you find the ratio of successive terms, you'll always get this common value. To find the common difference, I have to subtract a successive pair of terms.

It doesn't matter which pair I pick, as long as they're right next to each other. To be thorough, I'll do all the subtractions:. They gave me five terms, so the sixth term of the sequence is going to be the very next term. I find the next term by adding the common difference to the fifth term:.

To find the common ratio, I have to divide a successive pair of terms. To be thorough, I'll do all the divisions:.

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They gave me five terms, so the sixth term is the very next term; the seventh will be the term after that. To find the value of the seventh term, I'll multiply the fifth term by the common ratio twice:.

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Since arithmetic and geometric sequences are so nice and regular, they have formulas. For arithmetic sequences, the common difference is dand the first term a 1 is often referred to simply as " a ". Since we get the next term by adding the common difference, the value of a 2 is just:. At each stage, the common difference was multiplied by a value that was one less than the index. Following this pattern, the n -th term a n will have the form:. For geometric sequences, the common ratio is rand the first term a 1 is often referred to simply as " a ". Since we get the next term by multiplying by the common ratio, the value of a 2 is just:. At each stage, the common ratio was raised to a power that was one less than the index.

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## Geometric progression

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### Geometric series

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