Difference between revisions of "Geometric 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, $7, 14, 28, 56$ 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

Here are some problems that test knowledge of geometric sequences and series.

@@ Line 5: / Line 5: @@
 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>.
 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.
-===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>.
 '''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>.
 '''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.
 == Problems ==
+Here are some problems that test knowledge of geometric sequences and series.
 === Intermediate ===
 * [[2005_AIME_II_Problems/Problem_3 | 2005 AIME II Problem 3]]
 * [[2007 AIME II Problems/Problem 12 | 2007 AIME II Problem 12]]
-==See also==
+== See also ==
 *[[Arithmetic sequence]]
 *[[Sequence]]

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