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Getalpatrone

In vorige jare het jy patrone gesien in die vorm van prentjies en getalle. In hierde hoofstuk sal ons meer leer van die wiskunde van patrone. Patrone is herkenbaar as herhalende reekse wat gevind kan word in die natuur, vorms, gebeure, groepe van getalle en op baie ander plekke in ons daaglikse lewe. Byvoorbeeld, patrone kan gevind word in die sade van sonneblomme, sneeuvlokkies, geometriese patrone op lappieskomberse en teëls en reekse getalle soos 0; 4; 8; 12; 16; ...

Ondersoek : patrone

Kan jy die patrone herken in die volgende reekse van getalle?

  1. 2; 4; 6; 8; 10; ...
  2. 1; 2; 4; 7; 11; ...
  3. 1; 4; 9; 16; 25; ...
  4. 5; 10; 20; 40; 80; ...

Algemene getalpatrone

Reekse van getalle kan interessante patrone bevat. Die volgende is ʼn lys van die mees algemene patrone en hoe hulle gevorm word.

Voorbeelde:

  1. 1 ; 4 ; 7 ; 10 ; 13 ; 16 ; 19 ; 22 ; 25 ; . . . Hierdie reeks het ʼn verskil van 3 tussen al die getalle. Die patroon word gevorm deur elke keer 3 by te tel by die vorige getal.
  2. 3 ; 8 ; 13 ; 18 ; 23 ; 28 ; 33 ; 38 ; . . . Hierdie reeks het ʼn verskil van 5 tussen al die getalle. Die patroon word gevorm deur elke keer 5 by te tel by die vorige getal.
  3. 2 ; 4 ; 8 ; 16 ; 32 ; 64 ; 128 ; 256 ; . . . Hierdie reeks het ʼn faktor van 2 tussen al die getalle. Die volgende getal in die reeks word gevorm deur die vorige een met 2 te vermenigvuldig.
  4. 3 ; 9 ; 27 ; 81 ; 243 ; 729 ; 2187 ; . . . Hierdie reeks het ʼn faktor van 3 tussen al die getalle. Die volgende getal in die reeks word gevorm deur die vorige een met 3 te vermenigvuldig.

Spesiale reekse

Driehoeksgetalle

1 ; 3 ; 6 ; 10 ; 15 ; 21 ; 28 ; 36 ; 45 ; . . .

Hierdie reekse word gevorm deur ʼn patroon van kolletjies wat ʼn driehoek vorm. Deur nog ʼn ry kolletjies aan te heg (waar die elke nuwe ry een meer kolletjie bevat as die vorige een) en die kolletjies te tel, is dit moontlik om die volgende getal in die reeks te vind.

Vierkantsgetalle

1 ; 4 ; 9 ; 16 ; 25 ; 36 ; 49 ; 64 ; 81 ; . . .

Die waarde van ʼn term in die reeks word gevind deur die posisie (pleknommer in die ry) te kwadreer. Die tweede getal in die reeks is 2 kwadraat ( 2 2 o f 2 × 2 ). Die sewende getal is 7 kwadraat ( 7 2 o f 7 × 7 ) ens.

Derdemagsgetalle

1 ; 8 ; 27 ; 64 ; 125 ; 216 ; 343 ; 512 ; 729 ; . . .

Die waarde van ʼn term in die reeks word gevind deur die posisie tot die derde mag te verhef. Die tweede getal in die reeks is 2 tot die mag 3 ( 2 3 o f 2 × 2 × 2 ). Die sewende getal in die reeks is 7 tot die mag 3 ( 7 3 o f 7 × 7 × 7 ) ens.

Fibonacci getalle

0 ; 1 ; 1 ; 2 ; 3 ; 5 ; 8 ; 13 ; 21 ; 34 ; . . .

Die waarde van ʼn term in die reeks word gevind deur die vorige twee getalle in die reeks bymekaar te tel. Die 2 word gevind deur die vorige twee getalle in die reeks bymekaar te tel ( 1 + 1 ). Die 21 word gevind deur die twee getalle voor die 2 in die reeks bymekaar te tel ( 8 + 13 ). Die volgende getal in die reeks sal 55 wees ( 21 + 34 ).

Kan jy die volgende paar getalle vind?

Khan academy video oor getalpatrone - 1

Gestel jy en 3 vriende besluit om te studeer vir wiskunde, en dat julle om ʼn vierkantige tafel sit. ʼn Paar minute later sluit 2 ander vriende by julle aan en hulle wil kom sit. Om sitplek te kry vir hulle, besluit julle om ʼn tafel te skuif en dit langs julle tafel te sit sodat daar genoeg sitplek is vir die 6 van julle. Daarna besluit nog 2 van jou vriende om by julle aan te sluit en julle skuif ʼn derde tafel sodat daar genoeg plek is vir 8 van julle.

Twee ekstra mense kan sit vir elke tafel wat hulle bysit.

Ondersoek hoe die aantal mense om die tafels verband hou met die aantal tafels.

  1. Aantal tafels , n Aantal mense wat sitplek het
    1 4 = 4
    2 4 + 2 = 6
    3 4 + 2 + 2 = 8
    4 4 + 2 + 2 + 2 = 10
    n 4 + 2 + 2 + 2 + ... + 2
  2. Ons kan sien dat met 3 tafels is daar plek vir 8 mense, met 4 tafels is daar plek vir 10 mense ens. Ons begin met 4 mense en voeg elke keer 2 mense by. So, vir elke tafel wat bygevoeg word, is daar sitplek vir nog 2 mense.

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
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Devang Reply
are you nano engineer ?
s.
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Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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what's the easiest and fastest way to the synthesize AgNP?
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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I'm interested in nanotube
Uday
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what is nano technology
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
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name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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Damian
silver nanoparticles could handle the job?
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not now but maybe in future only AgNP maybe any other nanomaterials
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I'm interested in Nanotube
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this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Siyavula textbooks: wiskunde (graad 10) [caps]. OpenStax CNX. Aug 04, 2011 Download for free at http://cnx.org/content/col11328/1.4
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