Geometric sequence

A geometric sequence is a regular mathematical sequence of numbers with the property that the quotient of two adjacent sequence members is constant.

Origin of name

The term “geometric sequence” is derived from the geometric mean . For each member of a geometric sequence is the geometric mean of its neighboring members.

The summation of the following terms gives the geometric series .

Mathematical formulation

The -th member of a geometric sequence with the quotient is calculated from the formula ${\displaystyle i}$ ${\displaystyle a_{i}}$ ${\displaystyle q}$

${\displaystyle a_{i}=a_{1}\cdot q^{i-1}}$,

if the initial term is marked with, or ${\displaystyle a_{1}}$

${\displaystyle a_{i}=a_{0}\cdot q^{i}}$,

if the initial term is indicated with. ${\displaystyle a_{0}}$

The terms of a geometric sequence can also be calculated from the previous term using the following recursive formula:

${\displaystyle a_{i+1}=a_{i}\cdot q}$

Note : Every geometric sequence can be described with such a function rule, but such a function rule does not always describe a geometric sequence. The first term of a geometric sequence cannot be like this, because because of the prohibition of division by , the quotient of the first two terms of sequence does not exist for . Thus, the finite (from two members existing) sequences with the only geometrical sequences in which the number occurs as a result member or for which the number of the same is. In particular, there are no infinite geometric sequences with or with . ${\displaystyle a_{0}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\tfrac {a_{1}}{a_{0}}}}$ ${\displaystyle a_{0}=0}$ ${\displaystyle (a_{0},0)}$ ${\displaystyle a_{0}\neq 0}$ ${\displaystyle 0}$ ${\displaystyle q}$ ${\displaystyle 0}$ ${\displaystyle a_{i}=0}$ ${\displaystyle q=0}$

Numerical examples

example 1

The terms of the geometric sequence with the initial term and the quotient are: ${\displaystyle a_{0}=5}$ ${\displaystyle q=3}$

${\displaystyle a_{0}=5,\ a_{1}=15,\ a_{2}=45,\ a_{3}=135,\ \dotsc }$

If you just write the links one after the other, you get:

${\displaystyle 5,\ 15,\ 45,\ 135,\ 405,\ 1215,\ 3645,\ 10935,\ 32805,\ \dotsc }$

Example 2

The terms of the geometric sequence with the initial term and the quotient are: ${\displaystyle a_{0}=1}$ ${\displaystyle q=-{\tfrac {1}{2}}}$

${\displaystyle a_{0}=1,\ a_{1}=-{\frac {1}{2}},\ a_{2}={\frac {1}{4}},\ a_{3}=-{\frac {1}{8}},\ \dotsc }$

If you just write the links one after the other, you get:

${\displaystyle +1,\ -{\frac {1}{2}},\ +{\frac {1}{4}},\ -{\frac {1}{8}},\ +{\frac {1}{16}},\ -{\frac {1}{32}},\ +{\frac {1}{64}},\ -{\frac {1}{128}},\ +{\frac {1}{256}},\ \dotsc }$

Application examples

The geometric sequence describes growth processes in which the measured variable at the point in time results from the measured variable at the point in time by multiplying it by a constant factor . Examples: ${\displaystyle n+1}$ ${\displaystyle n}$ ${\displaystyle q}$

compound interest

→ Main article : Compound interest

At an interest rate of 5 percent , the capital increases every year by a factor of 1.05. So it is a geometric sequence with the ratio . The number here is called the interest factor . With a starting capital of 1000 euros this results ${\displaystyle q=1{,}05}$ ${\displaystyle q}$

• after one year a capital of

${\displaystyle 1000\,{\text{Euro}}\cdot 1{,}05=1050\,{\text{Euro}},}$

• after two years a capital of

${\displaystyle 1000\,{\text{Euro}}\cdot 1{,}05^{2}=1102{,}50\,{\text{Euro}},}$

• capital after three years

${\displaystyle 1000\,{\text{Euro}}\cdot 1{,}05^{3}=1157{,}63\,{\text{Euro}}}$

and so on.

Equal mood

→ Main article : Equal mood

There are several ways a musical instrument can be tuned. One of them is the equal mood . With it, the frequency ratio between two adjacent tones is always constant. If there are twelve notes in the octave, the sequence is:

${\displaystyle f(i)=a_{0}\cdot \left({\sqrt[{12}]{2}}\right)^{i}}$,

where, for example, is the frequency of the concert pitch and the semitone step distance to the concert pitch. is then the frequency of the tone sought with a semitone spacing from the "original tone " . ${\displaystyle a_{0}}$ ${\displaystyle i}$ ${\displaystyle f(i)}$ ${\displaystyle i}$ ${\displaystyle a_{0}}$

So the growth factor is . ${\displaystyle q={\sqrt[{12}]{2}}}$

Convergence of geometric sequences

An infinite geometric sequence is a zero sequence if and only if the absolute value of the real (or complex ) quotient of adjacent successor elements is less than 1. ${\displaystyle (a_{i})}$ ${\displaystyle |q|}$ ${\displaystyle q={\tfrac {a_{i+1}}{a_{i}}}}$

A. Assertion: is at least a null sequence if is. ${\displaystyle (a_{i})}$ ${\displaystyle |q|<1}$

Proof: be given. Claiming the existence of a with the property that for all the following applies: . ${\displaystyle \varepsilon >0}$ ${\displaystyle i_{0}}$ ${\displaystyle i>i_{0}}$ ${\displaystyle |a_{i}|<\varepsilon }$ ${\displaystyle \mathbf {(1)} }$

Because and exists ${\displaystyle 0<|q|<1}$ ${\displaystyle {\tfrac {\varepsilon }{|a_{0}|}}>0}$

${\displaystyle i_{0}=:{\frac {\ln \left({\frac {\varepsilon }{|a_{0}|}}\right)}{\ln(|q|)}}}$.

Where is the natural logarithm . ${\displaystyle \ln }$

Because , after multiplication by, the inequality sign is reversed for all : ${\displaystyle |q|<1\Rightarrow \ln(|q|)<0}$ ${\displaystyle i>i_{0}}$ ${\displaystyle \ln(|q|)}$

${\displaystyle i\cdot \ln |q|<i_{0}\cdot \ln |q|=\ln \left({\frac {\varepsilon }{|a_{0}|}}\right)}$;

for is ; Exponentiation (to the base ) does not change the inequality sign : ${\displaystyle i\in \mathbb {N} }$ ${\displaystyle i\cdot \ln |q|=\ln \left(|q|^{i}\right)=\ln \left(\left|q^{i}\right|\right)}$ ${\displaystyle e}$

${\displaystyle \left|q^{i}\right|<{\frac {\varepsilon }{|a_{0}|}}}$;

Because of this , the inequality sign remains unchanged after multiplication with the denominator; with : ${\displaystyle |a_{0}|>0}$ ${\displaystyle |a_{0}|\cdot \left|q^{i}\right|=\left|a_{0}\cdot q^{i}\right|}$

${\displaystyle \left|a_{0}\cdot q^{i}\right|<\varepsilon }$; hence (1), q. e. d.

B. Claim: is at most a null sequence if is. is not a null sequence if is. ${\displaystyle (a_{i})}$ ${\displaystyle |q|<1}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle (a_{i})}$ ${\displaystyle |q|\geq 1}$

Proof is (already) then no zero sequence when a can be selected such that for all the following applies: . ${\displaystyle (a_{i})}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle a_{i}}$ ${\displaystyle |a_{i}|\geq \varepsilon }$

Multiplication of the condition with results (because there is no inversion of the inequality sign ): ${\displaystyle |q|\geq 1}$ ${\displaystyle \left|a_{0}\cdot q^{i}\right|}$ ${\displaystyle \left|a_{0}\cdot q^{i}\right|>0}$

${\displaystyle \left|a_{0}\cdot q^{i+1}\right|\geq \left|a_{0}\cdot q^{i}\right|}$, in order to:

${\displaystyle |a_{i+1}|\geq |a_{i}|}$. . ${\displaystyle \mathbf {(2)} }$

One with is chosen. With (2) then applies to all : q. e. d. ${\displaystyle \varepsilon }$ ${\displaystyle |a_{0}|\geq \varepsilon >0}$ ${\displaystyle i>0}$ ${\displaystyle |a_{i}|\geq \varepsilon }$

See also

• Arithmetic sequence
• Growth factor (math)
• exponential process
• E series
• Renard series

Web links

• Explanations suitable for students

Bibliography

1. ^ Sequences and series . Retrieved March 14, 2010.
2. ↑ Eric W. Weisstein: Geometric Sequence. MathWorld , accessed November 10, 2019 .