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Difference between revisions of "Geometric sequence"
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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>.
  
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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]]. 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===

Revision as of 20:03, 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 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 has value $\frac{a_1}{1-r}$.

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

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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