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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 ?

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 ; ...

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

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

2 - 1 = 1 3 - 2 = 1 4 - 3 = 1 ...

Ons sien dan dat die tweede verskille gelyk is aan "1". Dus is [link] '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 ; ... }, wat gevorm is die die verskille tussen opeenvolgende terme van [link] te neem, 'n linêere formule van die vorm a x + b .

Kwadratiese rye

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 ; ...

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 term in die ry 5 , 12 , 23 , 38 , . . . , . . . ,

  1. i.e. 7 , 11 , 15

  2. die tweede verskil is 4.

    As ons die ry voortsit, sal die verskille tussen terme die volgende wees:

    15 + 4 = 19

    19 + 4 = 23

  3. Dus sal die volgende twee terme in die reeks die volgende wees:

    38 + 19 = 57

    57 + 23 = 80

    Dus sal die ry die volgende wees: 5 , 12 , 23 , 38 , 57 , 80

  4. Ons weet dat die tweede verskil 4 is. Die begin van die formule sal dus 2 n 2 wees.

  5. Indien n = 1 , moet jy die volgende waarde in die ry kry, wat "5" vir hierdie spesifieke ry is. Die verskil tussen 2 n 2 = 2 en die oorspronklike getal (5) is 3, wat lei tot n + 2 .

    Kyk of dit werk vir die tweede terme, d.i. wanneer n = 2 .

    Dan is 2 n 2 = 8 . Die verskil tussen term twee en (12) en 8 is 4, wat geskryf kan word as n + 2 .

    Dus vir die ry 5 , 12 , 23 , 38 , . . . is die formule vir die n de term 2 n 2 + n + 2 .

Algemene geval

Indien die ry kwadraties is, moet die n de term T n = a n 2 + b n + c wees

TERME a + b + c 4 a + 2 b + c 9 a + 3 b + c
1 ste verskil 3 a + b 5 a + b 7 a + b
2 de verskil 2 a 2 a

In elke geval is die tweede verskil 2 a . Hierdie feit kan gebruik word om a te vind, dan b en dan c .

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

  1. TERME 8 22 42 68
    1 ste verskil 14 20 26
    2 de verskil 6 6 6
  2. Dan is 2 a = 6 wat gee a = 3 En 3 a + b = 14 9 + b = 14 b = 5 En a + b + c = 8 3 + 5 + c = 8 c = 0
  3. Die formule is dus:     n de t e r m = 3 n 2 + 5 n

  4. Vir

    n = 1 , T 1 = 3 ( 1 ) 2 + 5 ( 1 ) = 8 n = 2 , T 2 = 3 ( 2 ) 2 + 5 ( 2 ) = 22 n = 3 , T 3 = 3 ( 3 ) 2 + 5 ( 3 ) = 42

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

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

a n = A · n 2 + B · n + C

waar A , B and C 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
Laat d = a 2 - a 1 d = 3 A + B
B = d - 3 A

Die gemene tweede verskil word gekry vanaf

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

Dus, vanuit [link] ,

B = d - 3 2 · D

Vanuit [link] ,

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

Uiteindelik word die algemene formule vir die n d e term van 'n kwadratiese ry gegee deur

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

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 100 ste term van die ry wees?
  1. Die getalle vermeerder met veelvoude van 6

    1 + 6 ( 1 ) = 7 , dan is 7 + 6 ( 2 ) = 19

    19 + 6 ( 3 ) = 37 , dan is 37 + 6 ( 4 ) = 61

    Dus is 61 + 6 ( 5 ) = 91

    Die volgende getal in die ry is 91.

  2. TERME 1 7 19 37 61
    1 ste verskil 6 12 18 24
    2 de verskil 6 6 6 6

    Die patroon sal 'n kwadratiese patroon opbring, aangesien die tweede verskille konstant is.

    Dus is a n 2 + b n + c = y

    Vir die eerste term: n = 1 , dan is y = 1

    Vir die tweede term: n = 2 , dan is y = 7

    Vir die derde term: n = 3 , dan is y = 19

    ensovoorts....

  3. a + b + c = 1 4 a + 2 b + c = 7 9 a + 3 b + c = 19
  4. verg. ( 2 ) - verg. ( 1 ) : 3 a + b = 6 verg. ( 3 ) - verg. ( 2 ) : 5 a + b = 12 verg. ( 5 ) - verg. ( 4 ) : 2 a = 6 a = 3 , b = - 3 e n c = 1
  5. Die algemene formule vir die patroon is 3 n 2 - 3 n + 1

  6. Vervang n met 100:

    3 ( 100 ) 2 - 3 ( 100 ) + 1 = 29 701

    Die waarde van die 100 ste term is 29 701.

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

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

gegee die kwadratiese ry

3 ; 6 ; 10 ; 15 ; 21 ; ...

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:
    a n = n 2 + 2 n + 1
  2. Bepaal watter van die volgende rye kwadraties is deur die gemene tweede verskille te bereken:
    1. 6 ; 9 ; 14 ; 21 ; 30 ; ...
    2. 1 ; 7 ; 17 ; 31 ; 49 ; ...
    3. 8 ; 17 ; 32 ; 53 ; 80 ; ...
    4. 9 ; 26 ; 51 ; 84 ; 125 ; ...
    5. 2 ; 20 ; 50 ; 92 ; 146 ; ...
    6. 5 ; 19 ; 41 ; 71 ; 109 ; ...
    7. 2 ; 6 ; 10 ; 14 ; 18 ; ...
    8. 3 ; 9 ; 15 ; 21 ; 27 ; ...
    9. 10 ; 24 ; 44 ; 70 ; 102 ; ...
    10. 1 ; 2 , 5 ; 5 ; 8 , 5 ; 13 ; ...
    11. 2 , 5 ; 6 ; 10 , 5 ; 16 ; 22 , 5 ; ...
    12. 0 , 5 ; 9 ; 20 , 5 ; 35 ; 52 , 5 ; ...
  3. Gegee a n = 2 n 2 , vind die waarde van n , a n = 242
  4. Gegee a n = ( n - 4 ) 2 , vind vir watter waarde van n , a n = 36
  5. Gegee a n = n 2 + 4 , vind die waarde van n , a n = 85
  6. Gegee a n = 3 n 2 , vind a 11
  7. Gegee a n = 7 n 2 + 4 n , vind a 9
  8. Gegee a n = 4 n 2 + 3 n - 1 , vind a 5
  9. Gegee a n = 1 , 5 n 2 , vind a 10
  10. Vir elke van die kwadratiese rye, vind die gemene tweede verskil, die formule vir die algemene term en gebruik dan die formule om a 100 te vind.
    1. 4 , 7 , 12 , 19 , 28 , ...
    2. 2 , 8 , 18 , 32 , 50 , ...
    3. 7 , 13 , 23 , 37 , 55 , ...
    4. 5 , 14 , 29 , 50 , 77 , ...
    5. 7 , 22 , 47 , 82 , 127 , ...
    6. 3 , 10 , 21 , 36 , 55 , ...
    7. 3 , 7 , 13 , 21 , 31 , ...
    8. 3 , 9 , 17 , 27 , 39 , ...

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Source:  OpenStax, Siyavula textbooks: wiskunde (graad 11). OpenStax CNX. Sep 20, 2011 Download for free at http://cnx.org/content/col11339/1.4
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