Difference between revisions of "Geometric sequence"
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More formally, the sequence <math>a_1, a_2, \ldots , a_n</math> is a geometric progression if and only if <math>a_2 / a_1 = a_3 / a_2 = \cdots = a_n / a_{n-1}</math>. This definition appears most frequently in its three-term form: namely, that constants <math>a</math>, <math>b</math>, and <math>c</math> are in geometric progression if and only if <math>b / a = c / b</math>. | More formally, the sequence <math>a_1, a_2, \ldots , a_n</math> is a geometric progression if and only if <math>a_2 / a_1 = a_3 / a_2 = \cdots = a_n / a_{n-1}</math>. This definition appears most frequently in its three-term form: namely, that constants <math>a</math>, <math>b</math>, and <math>c</math> are in geometric progression if and only if <math>b / a = c / b</math>. | ||
− | ==Properties== | + | == Properties == |
Because each term is a common multiple of the one before it, every term of a geometric sequence can be expressed as the sum of the first term and a multiple of the common ratio. Let <math>a_1</math> be the first term, <math>a_n</math> be the <math>n</math>th term, and <math>r</math> be the common ratio of any geometric sequence; then, <math>a_n = a_1 r^{n-1}</math>. | Because each term is a common multiple of the one before it, every term of a geometric sequence can be expressed as the sum of the first term and a multiple of the common ratio. Let <math>a_1</math> be the first term, <math>a_n</math> be the <math>n</math>th term, and <math>r</math> be the common ratio of any geometric sequence; then, <math>a_n = a_1 r^{n-1}</math>. | ||
A common lemma is that for any consecutive terms <math>a_{n-1}</math>, <math>a_n</math>, and <math>a_{n+1}</math> of a geometric sequence, then <math>a_n</math> is the [[geometric mean]] of <math>a_{n-1}</math> and <math>a_{n+1}</math>. In symbols, <math>a_n^2 = a_{n-1}a_{n+1}</math>. This is mostly used to perform substitutions. | A common lemma is that for any consecutive terms <math>a_{n-1}</math>, <math>a_n</math>, and <math>a_{n+1}</math> of a geometric sequence, then <math>a_n</math> is the [[geometric mean]] of <math>a_{n-1}</math> and <math>a_{n+1}</math>. In symbols, <math>a_n^2 = a_{n-1}a_{n+1}</math>. This is mostly used to perform substitutions. | ||
− | ==Sum== | + | == Sum == |
A '''geometric series''' is the sum of all the terms of a geometric sequence. They come in two varieties, both of which have their own formulas: finitely or infinitely many terms. | A '''geometric series''' is the sum of all the terms of a geometric sequence. They come in two varieties, both of which have their own formulas: finitely or infinitely many terms. | ||
− | ===Finite=== | + | === Finite === |
A finite geometric series with first term <math>a_1</math>, common ratio <math>r</math> not equal to one, and <math>n</math> total terms has a value equal to <math>\frac{a_1(r^n-1)}{r-1}</math>. | A finite geometric series with first term <math>a_1</math>, common ratio <math>r</math> not equal to one, and <math>n</math> total terms has a value equal to <math>\frac{a_1(r^n-1)}{r-1}</math>. | ||
'''Proof''': Let the geometric series have value <math>S</math>. Then <cmath>S = a_1 + a_1r + a_1r^2 + \cdots + a_1r^{n-1}.</cmath> Factoring out <math>a_1</math>, mulltiplying both sides by <math>(r-1)</math>, and using the [[Sum and difference of powers | difference of powers]] factorization yields <cmath>S(r-1) = a_1(r-1)(1 + r + r^2 + \cdots + r^{n-1}) = a_1(r^n-1).</cmath> Dividing both sides by <math>r-1</math> yields <math>S=\frac{a_1(r^n-1)}{r-1}</math>, as desired. <math>\square</math> | '''Proof''': Let the geometric series have value <math>S</math>. Then <cmath>S = a_1 + a_1r + a_1r^2 + \cdots + a_1r^{n-1}.</cmath> Factoring out <math>a_1</math>, mulltiplying both sides by <math>(r-1)</math>, and using the [[Sum and difference of powers | difference of powers]] factorization yields <cmath>S(r-1) = a_1(r-1)(1 + r + r^2 + \cdots + r^{n-1}) = a_1(r^n-1).</cmath> Dividing both sides by <math>r-1</math> yields <math>S=\frac{a_1(r^n-1)}{r-1}</math>, as desired. <math>\square</math> | ||
− | ===Infinite === | + | === Infinite === |
An infinite geometric series converges if and only if <math>|r|<1</math>; if this condition is satisfied, the series has value <math>\frac{a_1}{1-r}</math>. | An infinite geometric series converges if and only if <math>|r|<1</math>; if this condition is satisfied, the series has value <math>\frac{a_1}{1-r}</math>. | ||
'''Proof''': The proof that the series convergence if and only if <math>|r|<1</math> is an easy application of the ratio test from calculus; thus, such a proof is beyond the scope of this article. If one assumes convergence, there is an elementary proof of the formula that uses [[telescoping]]. Using the terms defined above, <cmath>S = a_1 + a_1r + a_1r^2 + \cdots.</cmath> Multiplying both sides by <math>r</math> and adding <math>a_1</math>, we find that <cmath>rS + a_1 = a_1 + r(a_1 + a_1r + \cdots) = a_1 + a_1r + a_1r^2 + \cdots = S.</cmath> Thus, <math>rS + a_1 = S</math>, and so <math>S = \frac{a_1}{1-r}</math>. <math>\square</math> | '''Proof''': The proof that the series convergence if and only if <math>|r|<1</math> is an easy application of the ratio test from calculus; thus, such a proof is beyond the scope of this article. If one assumes convergence, there is an elementary proof of the formula that uses [[telescoping]]. Using the terms defined above, <cmath>S = a_1 + a_1r + a_1r^2 + \cdots.</cmath> Multiplying both sides by <math>r</math> and adding <math>a_1</math>, we find that <cmath>rS + a_1 = a_1 + r(a_1 + a_1r + \cdots) = a_1 + a_1r + a_1r^2 + \cdots = S.</cmath> Thus, <math>rS + a_1 = S</math>, and so <math>S = \frac{a_1}{1-r}</math>. <math>\square</math> | ||
− | ===Common uses=== | + | === Common uses === |
One common instance of summing infinite geometric sequences is the [[decimal expansion]] of most [[rational number]]s. For instance, <math>0.33333\ldots = \frac 3{10} + \frac3{100} + \frac3{1000} + \frac3{10000} + \ldots</math> has first term <math>a_0 = \frac 3{10}</math> and common ratio <math>\frac1{10}</math>, so the infinite sum has value <math>S = \frac{\frac3{10}}{1-\frac1{10}} = \frac13</math>, just as we would have expected. | One common instance of summing infinite geometric sequences is the [[decimal expansion]] of most [[rational number]]s. For instance, <math>0.33333\ldots = \frac 3{10} + \frac3{100} + \frac3{1000} + \frac3{10000} + \ldots</math> has first term <math>a_0 = \frac 3{10}</math> and common ratio <math>\frac1{10}</math>, so the infinite sum has value <math>S = \frac{\frac3{10}}{1-\frac1{10}} = \frac13</math>, just as we would have expected. | ||
== Problems == | == Problems == | ||
+ | Here are some problems that test knowledge of geometric sequences and series. | ||
+ | |||
=== Intermediate === | === Intermediate === | ||
* [[2005_AIME_II_Problems/Problem_3 | 2005 AIME II Problem 3]] | * [[2005_AIME_II_Problems/Problem_3 | 2005 AIME II Problem 3]] | ||
* [[2007 AIME II Problems/Problem 12 | 2007 AIME II Problem 12]] | * [[2007 AIME II Problems/Problem 12 | 2007 AIME II Problem 12]] | ||
− | ==See also== | + | == See also == |
*[[Arithmetic sequence]] | *[[Arithmetic sequence]] | ||
*[[Sequence]] | *[[Sequence]] |
Revision as of 20:05, 3 November 2021
In algebra, a geometric sequence, sometimes called a geometric progression, is a sequence of numbers such that the ratio between any two consecutive terms is constant. This constant is called the common ratio of the sequence.
For example, is a geometric sequence with common ratio and is a geometric sequence with common ratio ; however, and are not geometric sequences, as the ratio between consecutive terms varies.
More formally, the sequence is a geometric progression if and only if . This definition appears most frequently in its three-term form: namely, that constants , , and are in geometric progression if and only if .
Contents
[hide]Properties
Because each term is a common multiple of the one before it, every term of a geometric sequence can be expressed as the sum of the first term and a multiple of the common ratio. Let be the first term, be the th term, and be the common ratio of any geometric sequence; then, .
A common lemma is that for any consecutive terms , , and of a geometric sequence, then is the geometric mean of and . In symbols, . This is mostly used to perform substitutions.
Sum
A geometric series is the sum of all the terms of a geometric sequence. They come in two varieties, both of which have their own formulas: finitely or infinitely many terms.
Finite
A finite geometric series with first term , common ratio not equal to one, and total terms has a value equal to .
Proof: Let the geometric series have value . Then Factoring out , mulltiplying both sides by , and using the difference of powers factorization yields Dividing both sides by yields , as desired.
Infinite
An infinite geometric series converges if and only if ; if this condition is satisfied, the series has value .
Proof: The proof that the series convergence if and only if is an easy application of the ratio test from calculus; thus, such a proof is beyond the scope of this article. If one assumes convergence, there is an elementary proof of the formula that uses telescoping. Using the terms defined above, Multiplying both sides by and adding , we find that Thus, , and so .
Common uses
One common instance of summing infinite geometric sequences is the decimal expansion of most rational numbers. For instance, has first term and common ratio , so the infinite sum has value , just as we would have expected.
Problems
Here are some problems that test knowledge of geometric sequences and series.