Difference between revisions of "Geometric sequence"
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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>. | 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. | + | 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 series''' is the sum of all the terms of a geometric sequence. They come in two varieties, both of which have their own formulas: finite and infinite. |
===Finite=== | ===Finite=== |
Revision as of 19:45, 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, 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 . This definition appears most frequently in its three-term form: namely, that constants , , and are in geometric progression if and only if .
Contents
[hide]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 for any consecutive terms , , and of a geometric sequence, then is the geometric mean of and . In symbols, . 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: finite and infinite.
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 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 . We say that the sum of the terms of this sequence is a convergent sum.
For instance, the series , sums to 2. The general formula for the sum of such a sequence is:
Where is the first term in the sequence, and is the common ratio.
Proof
Let the sequence be
Multiplying by yields,
We subtract these two equations to obtain:
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
thus,
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, has first term and common ratio , so the infinite sum has value , just as we would have expected.