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Difference between revisions of "Geometric sequence"

(→‎Summing a Geometric Sequence)
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A '''geometric sequence''' is a [[sequence]] of numbers in which each term is a fixed [[multiple]] of the previous term.  For example: 1, 2, 4, 8, 16, 32, ...  is a geometric sequence because each term is twice the previous term. In this case, 2 is called the ''common ratio'' of the sequence. More formally, a geometric sequence may be defined [[recursion|recursively]] by:
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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.
  
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<center><math>a_n = r\cdot a_{n-1}, n > 1</math></center>
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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.
  
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with a fixed first term <math>a_1</math> and common ratio <math>r</math>. Using this definition, the <math>n</math>th term has the closed-form:
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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>. 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>.
  
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<center><math>a_n = a_1\cdot r^{n-1}</math></center>
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== Properties ==
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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>.
  
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==Summing a Geometric Sequence==
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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.
  
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The sum of the first <math>n</math> terms of a geometric sequence is given by
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== Sum ==
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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.
  
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<center><math>S_n = a_1 + a_2 + \cdots + a_n = a_1\cdot\frac{r^n-1}{r-1}</math></center>
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=== Finite ===
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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>.
  
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where <math>a_1</math> is the first term in the sequence, and <math>r</math> is the common ratio.
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'''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>
  
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===Proof===
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=== Infinite ===
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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>.
  
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The geometric sequence can be rewritten as <math> a_1+r \cdot a_1+r^2 \cdot a_1+ \ldots + r^{n-1} \cdot a_1=a_1(1+r+r^2+ \ldots +r^{n-1})</math> where <math>n</math> is the amount of terms, <math>r</math> is the common ratio, and <math>a_1</math> is the first term. Multiplying in <math>(r-1)</math> yields <math>r^n-1</math> so <math> a_1 + a_2 + \ldots + a_n = a_1\cdot\frac{r^n-1}{r-1} </math>.
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'''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>
  
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==Infinite Geometric Sequences==
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== Problems ==
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Here are some problems with solutions that utilize geometric sequences and series.
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An [[infinite]] geometric sequence is a geometric sequence with an infinite number of terms.  If the common ratio is small, the terms will approach 0 and the sum of the terms will approach a fixed [[limit]].  In this case, "small" means <math>|r|<1</math>.  We say that the sum of the terms of this sequence is a [[convergent|convergent sum]].
 
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For instance, the series <math>1 + \frac12 + \frac14 + \frac18 + \cdots</math>, sums to 2.  The general formula for the sum of such a sequence is:
 
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<center><math>S = \frac{a_1}{1-r}</math>.</center> <br><br>
 
  
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Where <math>a_1</math> is the first term in the sequence, and <math>r</math> is the common ratio.
 
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===Proof===
 
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Let the sequence be
 
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<center><math>S=a_1+a_1r+a_1r^2+a_1r^3+\cdots</math></center>   
 
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Multiplying by <math>r</math> yields,
 
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<center><math>S \cdot r=a_1r+a_1r^2+a_1r^3+\cdots</math></center>
 
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We subtract these two equations to obtain:
 
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<center><math> S-Sr=a_1</math></center>
 
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There is only one term on the RHS because the rest of the terms cancel out after subtraction. Finally, we can factor and divide to get
 
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<center><math>S(1-r)=a_1</math></center>
 
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thus,
 
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<center><math>S=\frac{a_1}{1-r}</math></center>
 
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This method of multiplying the sequence and subtracting equations, called telescoping, is a frequently used method to evaluate infinite sequences.  In fact, the same method can be used to calculate the sum of a finite geometric sequence (given above).
 
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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.
 
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== Problems ==
 
 
=== 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]]
  
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==See also==
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== See also ==
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*[[Arithmetic sequence]]
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* [[Arithmetic sequence]]
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*[[Sequence]]
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* [[Harmonic sequence]]
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*[[Series]]
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* [[Sequence]]
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* [[Series]]
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[[Category:Algebra]] [[Category:Sequences and series]] [[Category:Definition]]

Latest revision as of 12:55, 7 March 2022

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, $1, 2, 4, 8$ is a geometric sequence with common ratio $2$ and $100, -50, 25, -25/2$ is a geometric sequence with common ratio $-1/2$; however, $1, 3, 9, -27$ and $-3, 1, 5, 9, \ldots$ are not geometric sequences, as the ratio between consecutive terms varies.

More formally, the sequence $a_1, a_2, \ldots , a_n$ is a geometric progression if and only if $a_2 / a_1 = a_3 / a_2 = \cdots = a_n / a_{n-1}$. A similar definition holds for infinite geometric sequences. It appears most frequently in its three-term form: namely, that constants $a$, $b$, and $c$ are in geometric progression if and only if $b / a = c / b$.

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 $a_1$ be the first term, $a_n$ be the $n$th term, and $r$ be the common ratio of any geometric sequence; then, $a_n = a_1 r^{n-1}$.

A common lemma is that a sequence is in geometric progression if and only if $a_n$ is the geometric mean of $a_{n-1}$ and $a_{n+1}$ for any consecutive terms $a_{n-1}, a_n, a_{n+1}$. In symbols, $a_n^2 = a_{n-1}a_{n+1}$. 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 $a_1$, common ratio $r$ not equal to one, and $n$ total terms has a value equal to $\frac{a_1(r^n-1)}{r-1}$.

Proof: Let the geometric series have value $S$. Then \[S = a_1 + a_1r + a_1r^2 + \cdots + a_1r^{n-1}.\] Factoring out $a_1$, mulltiplying both sides by $(r-1)$, and using the difference of powers factorization yields \[S(r-1) = a_1(r-1)(1 + r + r^2 + \cdots + r^{n-1}) = a_1(r^n-1).\] Dividing both sides by $r-1$ yields $S=\frac{a_1(r^n-1)}{r-1}$, as desired. $\square$

Infinite

An infinite geometric series converges if and only if $|r|<1$; if this condition is satisfied, the series converges to $\frac{a_1}{1-r}$.

Proof: The proof that the series convergence if and only if $|r|<1$ 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, \[S = a_1 + a_1r + a_1r^2 + \cdots.\] Multiplying both sides by $r$ and adding $a_1$, we find that \[rS + a_1 = a_1 + r(a_1 + a_1r + \cdots) = a_1 + a_1r + a_1r^2 + \cdots = S.\] Thus, $rS + a_1 = S$, and so $S = \frac{a_1}{1-r}$. $\square$

Problems

Here are some problems with solutions that utilize geometric sequences and series.

Intermediate

See also

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