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## Inleiding

In graad 10 het jy geleer van rekenkundige rye, waar die verskil tussen opeenvolgende terme konstant was. In hierdie hoofstuk leer ons van kwadratiese rye.

## Wat is 'n kwadratiese ry?

'n Kwadratiese ry is 'n ry waar die tweede verskille tussen opeenvolgende terme met dieselfde hoeveelheid verskil. Dit word 'n gemene tweede verskil genoem.

Byvoorbeeld

1 ; 2 ; 4 ; 7 ; 11 ; ... 1 ; 2 ; 4 ; 7 ; 11 ; ...
(1)

is 'n kwadratiese ry. Kom ons stel vas hoekom ...

Indien ons die verskil tussen opeenvolgende terme neem, is

a 2 - a 1 = 2 - 1 = 1 a 3 - a 2 = 4 - 2 = 2 a 4 - a 3 = 7 - 4 = 3 a 5 - a 4 = 11 - 7 = 4 a 2 - a 1 = 2 - 1 = 1 a 3 - a 2 = 4 - 2 = 2 a 4 - a 3 = 7 - 4 = 3 a 5 - a 4 = 11 - 7 = 4
(2)

dan werk ons die tweede verskille uit, wat bloot gekry word deur die verskille tussen opeenvolgende verskille {1;2;3;4;...1;2;3;4;...} te neem:

2 - 1 = 1 3 - 2 = 1 4 - 3 = 1 ... 2 - 1 = 1 3 - 2 = 1 4 - 3 = 1 ...
(3)

Ons sien dan dat die tweede verskille gelyk is aan "1". Dus is vergelyking 1 'n kwadratiese ry.

Let op dat die verskille tussen opeenvolgende terme (met ander woorde, die eerste verskille) van 'n kwadratiese ry, 'n ry vorm waar daar 'n konstante verskil is tussen opeenvolgende terme. In die voorbeeld hier bo, het die ry {1;2;3;4;...1;2;3;4;...}, wat gevorm is die die verskille tussen opeenvolgende terme van vergelyking 1 te neem, 'n linêere formule van die vorm ax+bax+b.

Die volgende is ook voorbeelde van kwadratiese rye:

3 ; 6 ; 10 ; 15 ; 21 ; ... 4 ; 9 ; 16 ; 25 ; 36 ; ... 7 ; 17 ; 31 ; 49 ; 71 ; ... 2 ; 10 ; 26 ; 50 ; 82 ; ... 31 ; 30 ; 27 ; 22 ; 15 ; ... 3 ; 6 ; 10 ; 15 ; 21 ; ... 4 ; 9 ; 16 ; 25 ; 36 ; ... 7 ; 17 ; 31 ; 49 ; 71 ; ... 2 ; 10 ; 26 ; 50 ; 82 ; ... 31 ; 30 ; 27 ; 22 ; 15 ; ...
(4)

Kan jy die gemene tweede verskille vir elk van die voorbeelde hier bo bereken?

Skryf neer die volgende twee terme en vind 'n formule vir die n de n de term in die ry 5,12,23,38,...,...,5,12,23,38,...,...,

### Algemene geval

Indien die ry kwadraties is, moet die n de n de term Tn=an2+bn+cTn=an2+bn+c wees

 TERME a + b + c a + b + c 4 a + 2 b + c 4 a + 2 b + c 9 a + 3 b + c 9 a + 3 b + c 1ste1ste verskil 3 a + b 3 a + b 5 a + b 5 a + b 7 a + b 7 a + b 2de2de verskil 2 a 2 a 2 a 2 a

In elke geval is die tweede verskil 2a2a. Hierdie feit kan gebruik word om aa te vind, dan bb en dan cc.

Die volgende ry is kwadraties: 8,22,42,68,...8,22,42,68,... Vind die formule.

#### Bepaling van die n de n de -term van 'n kwadratiese ry

Laat die ndende-term vir 'n kwadratiese ry gegee word deur

a n = A · n 2 + B · n + C a n = A · n 2 + B · n + C
(7)

waar AA, BB and CC konstantes is wat bepaal moet word.

a n = A · n 2 + B · n + C a 1 = A ( 1 ) 2 + B ( 1 ) + C = A + B + C a 2 = A ( 2 ) 2 + B ( 2 ) + C = 4 A + 2 B + C a 3 = A ( 3 ) 2 + B ( 3 ) + C = 9 A + 3 B + C a n = A · n 2 + B · n + C a 1 = A ( 1 ) 2 + B ( 1 ) + C = A + B + C a 2 = A ( 2 ) 2 + B ( 2 ) + C = 4 A + 2 B + C a 3 = A ( 3 ) 2 + B ( 3 ) + C = 9 A + 3 B + C
(8)
Laat d = a 2 - a 1 d = 3 A + B Laat d = a 2 - a 1 d = 3 A + B
(9)
B = d - 3 A B = d - 3 A
(10)

Die gemene tweede verskil word gekry vanaf

D = ( a 3 - a 2 ) - ( a 2 - a 1 ) = ( 5 A + B ) - ( 3 A + B ) = 2 A D = ( a 3 - a 2 ) - ( a 2 - a 1 ) = ( 5 A + B ) - ( 3 A + B ) = 2 A
(11)
A = D 2 A = D 2
(12)

Dus, vanuit vergelyking 10,

B = d - 3 2 · D B = d - 3 2 · D
(13)

Vanuit vergelyking 8,

C = a 1 - ( A + B ) = a 1 - D 2 - d + 3 2 · D C = a 1 - ( A + B ) = a 1 - D 2 - d + 3 2 · D
(14)
C = a 1 + D - d C = a 1 + D - d
(15)

Uiteindelik word die algemene formule vir die ndendeterm van 'n kwadratiese ry gegee deur

a n = D 2 · n 2 + ( d - 3 2 D ) · n + ( a 1 - d + D ) a n = D 2 · n 2 + ( d - 3 2 D ) · n + ( a 1 - d + D )
(16)

#### Exercise 3: Die gebruik van die stel vergelykings

Bestudeer die volgende patroon: 1; 7; 19; 37; 61; ...

1. Wat is die volgende getal in die ry?
2. Gebruik veranderlikes om 'n algebraïese formula op te stel wat die patroon veralgemeen.
3. Wat sal die 100ste100ste term van die ry wees?

#### Teken 'n grafiek van die terme van 'n kwadratiese ry

Die plot van anan vs. nn lewer 'n paraboliese grafiek vir 'n kwadratiese ry,

3 ; 6 ; 10 ; 15 ; 21 ; ... 3 ; 6 ; 10 ; 15 ; 21 ; ...
(19)

Indien ons elke van die terme teenoor die ooreenstemmende indeks teken, kry ons die grafiek van 'n parabool.

## Oefeninge

1. Vind die eerste 5 terme van die kwadratiese ry gedefinieer deur:
an=n2+2n+1an=n2+2n+1
(20)
2. Bepaal watter van die volgende rye kwadraties is deur die gemene tweede verskille te bereken:
1. 6;9;14;21;30;...6;9;14;21;30;...
2. 1;7;17;31;49;...1;7;17;31;49;...
3. 8;17;32;53;80;...8;17;32;53;80;...
4. 9;26;51;84;125;...9;26;51;84;125;...
5. 2;20;50;92;146;...2;20;50;92;146;...
6. 5;19;41;71;109;...5;19;41;71;109;...
7. 2;6;10;14;18;...2;6;10;14;18;...
8. 3;9;15;21;27;...3;9;15;21;27;...
9. 10;24;44;70;102;...10;24;44;70;102;...
10. 1;2,5;5;8,5;13;...1;2,5;5;8,5;13;...
11. 2,5;6;10,5;16;22,5;...2,5;6;10,5;16;22,5;...
12. 0,5;9;20,5;35;52,5;...0,5;9;20,5;35;52,5;...
3. Gegee an=2n2an=2n2, vind die waarde van nn, an=242an=242
4. Gegee an=(n-4)2an=(n-4)2, vind vir watter waarde van nn, an=36an=36
5. Gegee an=n2+4an=n2+4, vind die waarde van nn, an=85an=85
6. Gegee an=3n2an=3n2, vind a11a11
7. Gegee an=7n2+4nan=7n2+4n, vind a9a9
8. Gegee an=4n2+3n-1an=4n2+3n-1, vind a5a5
9. Gegee an=1,5n2an=1,5n2, vind a10a10
10. Vir elke van die kwadratiese rye, vind die gemene tweede verskil, die formule vir die algemene term en gebruik dan die formule om a100a100 te vind.
1. 4,7,12,19,28,...4,7,12,19,28,...
2. 2,8,18,32,50,...2,8,18,32,50,...
3. 7,13,23,37,55,...7,13,23,37,55,...
4. 5,14,29,50,77,...5,14,29,50,77,...
5. 7,22,47,82,127,...7,22,47,82,127,...
6. 3,10,21,36,55,...3,10,21,36,55,...
7. 3,7,13,21,31,...3,7,13,21,31,...
8. 3,9,17,27,39,...3,9,17,27,39,...

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