CZ:Featured article/Current: Difference between revisions

Revision as of 20:55, 26 July 2012

Geometric sequence

A geometric sequence (or geometric progression) is a (finite or infinite) sequence of (real or complex) numbers such that the quotient (or ratio) of consecutive elements is the same for every pair.

In finance, compound interest generates a geometric sequence.

Examples

Examples for geometric sequences are

$3,6,12,24,48,96$ (finite, length 6: 6 elements, quotient 2)

$1,-2,4,-8$ (finite, length 4: 4 elements, quotient −2)

$8,4,2,1,{1 \over 2},{1 \over 4},{1 \over 8},\dots {1 \over 2^{n-4}},\dots$ (infinite, quotient $1 \over 2$ )

$2,2,2,2,\dots$ (infinite, quotient 1)

$-2,2,-2,2,\dots ,(-1)^{n}\cdot 2,\dots$ (infinite, quotient −1)

${1 \over 2},1,2,4,\dots ,2^{n-2},\dots$ (infinite, quotient 2)

$1,0,0,0,\dots \$ (infinite, quotient 0) (See General form below)

Application in finance

The computation of compound interest leads to a geometric series:

When an initial amount A is deposited at an interest rate of p percent per time period then the value A_n of the deposit after n time-periods is given by

A_{n}=A\left(1+{p \over 100}\right)^{n}

i.e., the values A=A₀, A₁, A₂, A₃, ... form a geometric sequence with quotient q = 1+(p/100).

.... (read more)

notes

@@ Line 1: / Line 1: @@
-== '''[[Jet engine]]''' ==
+== '''[[Geometric sequence]]''' ==
 ----
-A '''jet engine''' is a reaction engine that discharges a fast moving jet of fluid to generate thrust in accordance with Newton's laws of motion. This broad definition of jet engines includes turbojets, turbofans, rockets, ramjets, pulse jets and pump-jets. In general, most jet engines are [[internal combustion engine]]s but non-combusting forms also exist.
-In some common usage, the term ''jet engine'' generally refers to an internal combustion duct engine, which typically consists of an engine with a rotary (rotating) air compressor powered by a turbine ("Brayton cycle"), with the leftover power providing thrust via a propelling nozzle. These types of jet engines are primarily used by jet aircraft for long distance travel. The early jet aircraft used turbojet engines which were relatively inefficient for subsonic flight. Modern subsonic jet aircraft usually use high-bypass turbofan engines which help give high speeds as well as, over long distances, giving better fuel efficiency than many other forms of transport.
+A '''geometric sequence''' (or '''geometric progression''') is a (finite or infinite) [[sequence]]
-{{Image|Jet engine.png|right|260px|A typical gas turbine jet engine (also known as a "turbojet"). Air is compressed by the fan blades as it enters the engine, and it is mixed and burned with fuel in the combustion section. The hot exhaust gases provide forward thrust and turn the turbines which drive the compressor fan blades.}}
+of (real or complex) numbers
-===History===
+such that the quotient (or ratio) of consecutive elements is the same for every pair.
-Jet engines can be dated back to the first century AD, when [[Hero of Alexandria]] (a [[Greek]] mathematician) invented the aeolipile. This used steam power directed through two jet nozzles so as to cause a sphere to spin rapidly on its axis. So far as is known, it was little used for supplying mechanical power, and the potential practical applications of Hero's invention of the jet engine were not recognized. It was simply considered a curiosity.
+In finance, compound [[interest rate|interest]] generates a geometric sequence.
-''[[Jet engine|.... (read more)]]''
+=== Examples ===
+Examples for geometric sequences are
+* <math> 3, 6, 12, 24, 48, 96            </math> (finite, length 6: 6 elements, quotient 2)
+* <math> 1, -2, 4, -8                    </math> (finite, length 4: 4 elements, quotient &minus;2)
+* <math> 8, 4, 2, 1, {1\over2}, {1\over4}, {1\over8},
+                     \dots {1\over2^{n-4}}, \dots  </math> (infinite, quotient <math>1\over2</math>)
+* <math> 2, 2, 2, 2, \dots </math> (infinite, quotient 1)
+* <math> -2, 2, -2, 2, \dots , (-1)^n\cdot 2 , \dots </math>  (infinite, quotient &minus;1)
+* <math> {1\over2}, 1, 2, 4, \dots , 2^{n-2}, \dots </math>  (infinite, quotient 2)
+* <math> 1, 0, 0, 0, \dots \ </math> (infinite, quotient 0) (See [[#General form|General form]] below)
+=== Application in finance ===
+The computation of compound interest leads to a geometric series:
+When an initial amount ''A'' is deposited at an interest rate of ''p'' percent per time period
+then the value ''A''<sub>n</sub> of the deposit after ''n'' time-periods is given by
+:: <math> A_n = A \left( 1 + {p\over100} \right)^n </math>
+i.e., the values
+''A''=''A''<sub>0</sub>, ''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>, ...
+form a geometric sequence with quotient ''q'' = 1+(''p''/100).
+''[[Geometric sequence|.... (read more)]]''
 {| class="wikitable collapsible collapsed" style="width: 90%; float: center; margin: 0.5em 1em 0.8em 0px;"
 |-
-! style="text-align: center;" | &nbsp;[[Jet engine#References|notes]]
+! style="text-align: center;" | &nbsp;[[Geometric sequence|notes]]
 |-
 |
 {{reflist|2}}
 |}

CZ:Featured article/Current: Difference between revisions

Revision as of 20:55, 26 July 2012

Geometric sequence

Examples

Application in finance

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