Difference between revisions of "Geometric sequence"
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− | + | 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, <math>1, 2, 4, 8</math> is a geometric sequence with common ratio <math>2</math> and <math>100, -50, 25, -25/2</math> is a geometric sequence with common ratio <math>-1/2</math>; however, <math>1, 3, 9, -27</math> and <math>-3, 1, 5, 9, \ldots</math> are not geometric sequences, as the ratio between consecutive terms varies. | |
− | <math>a_n = | + | 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>. A similar definition holds for infinite geometric sequences. It 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 == | |
+ | 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>. | ||
− | <math> | + | A common lemma is that a sequence is in geometric progression if and only if <math>a_n</math> is the [[geometric mean]] of <math>a_{n-1}</math> and <math>a_{n+1}</math> for any consecutive terms <math>a_{n-1}, a_n, a_{n+1}</math>. In symbols, <math>a_n^2 = a_{n-1}a_{n+1}</math>. This is mostly used to perform substitutions, though it occasionally serves as a definition of geometric sequences. |
− | == | + | == 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 <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>. | ||
− | <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 === | |
+ | An infinite geometric series converges if and only if <math>|r|<1</math>; if this condition is satisfied, the series [[Convergent | converges]] to <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 series|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> |
− | + | == Problems == | |
+ | Here are some problems with solutions that utilize geometric sequences and series. | ||
− | + | === Intermediate === | |
+ | * [[1965 AHSME Problems/Problem 36 | 1965 AHSME Problem 36]] | ||
+ | * [[2005_AIME_II_Problems/Problem_3 | 2005 AIME II Problem 3]] | ||
+ | * [[2007 AIME II Problems/Problem 12 | 2007 AIME II Problem 12]] | ||
− | + | == See also == | |
+ | * [[Arithmetic sequence]] | ||
+ | * [[Harmonic sequence]] | ||
+ | * [[Sequence]] | ||
+ | * [[Series]] | ||
− | + | [[Category:Algebra]] [[Category:Sequences and series]] [[Category:Definition]] | |
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Latest revision as of 21:09, 19 July 2024
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
. A similar definition holds for infinite geometric sequences. It appears most frequently in its three-term form: namely, that constants
,
, and
are in geometric progression if and only if
.
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 a sequence is in geometric progression if and only if is the geometric mean of
and
for any consecutive terms
. In symbols,
. This is mostly used to perform substitutions, though it occasionally serves as a definition of geometric sequences.
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 converges to
.
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
.
Problems
Here are some problems with solutions that utilize geometric sequences and series.