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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;...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;...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;...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;...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 ; . . . 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 ; . . . 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 (22of2×222of2×2
). Die sewende getal is 7 kwadraat (72of7×772of7×7
) ens.

Derdemagsgetalle

1 ; 8 ; 27 ; 64 ; 125 ; 216 ; 343 ; 512 ; 729 ; . . . 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 (23of2×2×223of2×2×2
). Die sewende getal in die reeks is 7 tot die mag 3 (73of7×7×773of7×7×7
) ens.

Fibonacci Getalle

0 ; 1 ; 1 ; 2 ; 3 ; 5 ; 8 ; 13 ; 21 ; 34 ; . . . 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+11+1). Die 21 word gevind deur die twee getalle voor die 2 in die reeks bymekaar te tel (8+138+13). Die volgende getal in die reeks sal 55 wees (21+3421+34).

Kan jy die volgende paar getalle vind?

Figure 1
Khan Academy video oor getalpatrone - 1

Exercise 1: Studeertafels

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.

Figure 2: Twee ekstra mense kan sit vir elke tafel wat hulle bysit.
Figure 2 (MG10C7_002.png)

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

Solution

  1. Step 1. Tabelleer ʼn paar terme om te sien of daar ʼn patroon is :
    Table 1
    Aantal tafels, nn Aantal mense wat sitplek het
    1 4 = 4 4 = 4
    2 4 + 2 = 6 4 + 2 = 6
    3 4 + 2 + 2 = 8 4 + 2 + 2 = 8
    4 4 + 2 + 2 + 2 = 10 4 + 2 + 2 + 2 = 10
    ⋮ ⋮ ⋮⋮
    n n 4 + 2 + 2 + 2 + ... + 2 4 + 2 + 2 + 2 + ... + 2
  2. Step 2. Beskryf die patroon :

    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.

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