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Difference between revisions of "Geometric sequence"
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==Properties==
 
==Properties==
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The <math>n</math>th term has the closed-form:
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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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<center><math>a_n = a_1\cdot r^{n-1}</math></center>
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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.
  
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==Summing a Geometric Sequence==
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==Sum==
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A '''geometric series''' is the sum of all the terms of an arithmetic sequence. They come in two varieties, which have their own formulas: finite and infinite.
  
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The sum of the first <math>n</math> terms of a geometric sequence is given by
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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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<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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'''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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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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The geometric sequence can be rewritten as <math> a_1+r \cdot a_1+r^2 \cdot a_1+ \cdots + r^{n-1} \cdot a_1=a_1(1+r+r^2+ \cdots +r^{n-1})</math> where <math>n</math> is the number of terms, <math>r</math> is the common ratio, and <math>a_1</math> is the first term. Multiplying by <math>(r-1)</math> yields <math>r^n-1</math> so <math> a_1 + a_2 + \cdots + a_n = a_1\cdot\frac{r^n-1}{r-1} </math>.
 
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==Infinite Geometric Sequences==
 
  
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===Infinite Geometric Sequences===
 
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]].
 
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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===Proof===  
 
===Proof===  
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Let the sequence be  
 
Let the sequence be  
  

Revision as of 16:51, 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, $7, 14, 28, 56$ is a geometric sequence with common ratio $2$ and $100, -50, 25, -25/2, \ldots$ is a geometric sequence with common ratio $-1/2$; however, $1, 3, 9, 19$ and $-3, 1, 5, 9$ 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 $a_2 / a_1 = a_3 / a_2 = \cdots = a_n / a_{n-1}$. This definition 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 for any consecutive terms $a_{n-1}$, $a_n$, and $a_{n+1}$ of a geometric sequence, then $a_n$ is the geometric mean of $a_{n-1}$ and $a_{n+1}$. In symbols, $a_n^2 = a_{n-1}a_{n+1}$. This is mostly used to perform substitutions.

Sum

A geometric series is the sum of all the terms of an arithmetic sequence. They come in two varieties, which have their own formulas: finite and infinite.

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 Geometric Sequences

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 $|r|<1$. We say that the sum of the terms of this sequence is a convergent sum.

For instance, the series $1 + \frac12 + \frac14 + \frac18 + \cdots$, sums to 2. The general formula for the sum of such a sequence is:

$S = \frac{a_1}{1-r}$.



Where $a_1$ is the first term in the sequence, and $r$ is the common ratio.

Proof

Let the sequence be

$S=a_1+a_1r+a_1r^2+a_1r^3+\cdots.$

Multiplying by $r$ yields,

$S \cdot r=a_1r+a_1r^2+a_1r^3+\cdots.$

We subtract these two equations to obtain:

$S-Sr=a_1.$

There is only one term on the right side of the equation because the rest of the terms cancel out after subtraction. Finally, we can factor and divide to get

$S(1-r)=a_1$

thus,

$S=\frac{a_1}{1-r}.$

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).

Common uses

One common instance of summing infinite geometric sequences is the decimal expansion of most rational numbers. For instance, $0.33333\ldots = \frac 3{10} + \frac3{100} + \frac3{1000} + \frac3{10000} + \ldots$ has first term $a_0 = \frac 3{10}$ and common ratio $\frac1{10}$, so the infinite sum has value $S = \frac{\frac3{10}}{1-\frac1{10}} = \frac13$, just as we would have expected.

Problems

Intermediate

See also

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