Geometric sequence

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: 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.

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Geometric sequence

Contents

Properties

Sum

Finite

Infinite Geometric Sequences

Proof

Common uses

Problems

Intermediate

See also