Imibuzo Eyisibonelo Exoxa Ngochungechunge Lwejiyomethri Olungapheli
Uchungechunge lwejometri olungenamkhawulo luwuchungechunge oluhlanganisa inani elingenamkhawulo lamagama ekuqhubekeni kwejometri. Lolu chungechunge lunezidingo ezithile zokubalwa kwesamba sawo (noma umkhawulo) sawo. Kulesi sihloko, sizoxoxa ngomqondo oyisisekelo wochungechunge lwejometri olungenamkhawulo, izidingo zokwakha, kanye nezinkinga eziningana zezibonelo kanye nezixazululo zazo.
Imiqondo Eyisisekelo Yochungechunge Lwejiyomethri Olungapheli
Empeleni, uchungechunge lwejiyomethri luwuchungechunge lwezinombolo lapho itemu ngalinye ngemva kwelokuqala litholakala ngokuphindaphinda itemu elidlule ngesilinganiso esivamile esibizwa ngokuthi isilinganiso esivamile (r). Ake sithi sinochungechunge lwejiyomethri:
\[ a, ar, ar^2, ar^3, ar^4, \ldots \]
Ngochungechunge lwejometri olungenamkhawulo, sibheka isamba sawo wonke amagama ochungechungeni. Isamba salolu chungechunge sichazwa kanje:
\[ S = a + ar + ar^2 + ar^3 + ar^4 + \ldots \]
Isamba sochungechunge lwejiyomethri olungenamkhawulo siyahlangana (sinenani eliqondile) uma futhi kuphela uma inani lesilinganiso \( |r| < 1 \). Uma \( |r| \geq 1 \), khona-ke uchungechunge luyahlukana futhi alunaso isamba esiqondile (siya ku-infinity). Uma \( |r| < 1 \), isamba S sochungechunge lwejiyomethri olungenamkhawulo singachazwa ngokuthi: \[ S = \frac{a}{1-r} \] lapho: - \( S \) kuyisamba sochungechunge, - \( a \) kuyigama lokuqala, - \( r \) kuyisilinganiso esivamile. Imibuzo Yesibonelo Nengxoxo Umbuzo 1: Thola isamba sochungechunge lwejiyometri olungenamkhawulo kulolu chungechunge olulandelayo: \[ 3 + 1.5 + 0.75 + 0.375 + \ldots \] Ingxoxo: Ake sithole izakhi ezibalulekile zochungechunge: Igama lokuqala \( a = 3 \) Isilinganiso \( r \) singatholakala ngokuhlukanisa igama lesibili ngegama lokuqala, okungukuthi: \[ r = \frac{1.5}{3} = 0.5 \] Njengoba \( |r| = 0.5 < 1 \), lolu chungechunge luyahlangana futhi singabala inani lochungechunge olungenamkhawulo. Sebenzisa ifomula yesamba sochungechunge lwejiyomethri olungenamkhawulo: \[ S = \frac{a}{1-r} \] \[ S = \frac{3}{1-0.5} \] \[ S = \frac{3}{0.5} \] \[ S = 6 \] Ngakho-ke, isamba sochungechunge lwejiyomethri olungenamkhawulo singu-6. Isibonelo Umbuzo 2 Umbuzo: Thola isamba sochungechunge lwejiyomethri olungenamkhawulo nethemu yokuqala engu-8 kanye nesilinganiso \( r = -\frac{1}{3} \). Isixazululo: Ithemu yokuqala \( a = 8 \) Isilinganiso \( r = -\frac{1}{3} \) Njengoba \( |r| = \frac{1}{3} < 1 \), lolu chungechunge luyahlangana futhi singabala isamba sochungechunge olungenamkhawulo. Sebenzisa ifomula yesamba sochungechunge lwejiyometri olungenamkhawulo: \[ S = \frac{a}{1-r} \] \[ S = \frac{8}{1 - \left(-\frac{1}{3}\right)} \] \[ S = \frac{8}{1 + \frac{1}{3}} \] \[ S = \frac{8}{\frac{4}{3}} \] \[ S = 8 \times \frac{3}{4} \] \[ S = 6 \] Ngakho-ke, isamba sochungechunge lwejiyometri olungenamkhawulo singu-6. Isibonelo Umbuzo 3 Umbuzo: Ingabe uchungechunge olulandelayo lunesamba esingenamkhawulo? Uma kunjalo, thola isamba. \[ 5 + 2.5 + 1.25 + 0.625 + \ldots \] Ingxoxo: Ithemu lokuqala \( a = 5 \) Isilinganiso \( r \) singatholakala ngokuhlukanisa ithemu lesibili ngethemu lokuqala, okungukuthi: \[ r = \frac{2.5}{5} = 0.5 \] Njengoba \( |r| = 0.5 < 1 \), lolu chungechunge luyahlangana futhi singabala isamba sochungechunge olungenamkhawulo. Sebenzisa ifomula yesamba sochungechunge lwejiyomethri olungenamkhawulo: \[ S = \frac{a}{1-r} \] \[ S = \frac{5}{1-0.5} \] \[ S = \frac{5}{0.5} \] \[ S = 10 \] Ngakho-ke, isamba sochungechunge lwejiyomethri olungenamkhawulo ngu-10. Isibonelo Umbuzo 4 Umbuzo: Nquma ukuthi uchungechunge olulandelayo luhlangana noma luhlukene: \[ 4 - 6 + 9 - 13.5 + \ldots \] Isixazululo: Ithemu yokuqala \( a = 4 \) Isilinganiso \( r \) singatholakala ngokuhlukanisa ithemu yesibili ngethemu yokuqala, okungukuthi: \[ r = \frac{-6}{4} = -1.5 \] Njengoba \( |r| = 1.5 > 1 \), lolu chungechunge luhlukile futhi alunaso isamba esiqondile.
Ngakho-ke, uchungechunge luhlukile.
Isibonelo Umbuzo 5
Umbuzo: Ake sithi unezinchungechunge ezilandelayo ezingenamkhawulo:
\[ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots \]
Nquma isamba sochungechunge.
Ingxoxo:
Ithemu lokuqala \( a = \frac{1}{2} \)
Isilinganiso \(r \) singatholakala ngokuhlukanisa itemu lesibili ngetemu lokuqala, okungukuthi:
\[ r = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2} \]
Njengoba \( |r| = \frac{1}{2} < 1 \), lolu chungechunge luyahlangana futhi singabala isamba sochungechunge olungenamkhawulo. Sebenzisa ifomula yesamba sochungechunge olungenamkhawulo lwejiyomethri: \[ S = \frac{a}{1-r} \] \[ S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} \] \[ S = \frac{\frac{1}{2}}{\frac{1}{2}} \] \[ S = 1 \] Ngakho-ke, isamba sochungechunge olungenamkhawulo lwejiyomethri ngu-1. Isiphetho Uchungechunge olungenamkhawulo lwejiyomethri luwumqondo obalulekile wezibalo onezinhlelo zokusebenza ezibanzi emikhakheni eyahlukene. Ukuze sinqume isamba sochungechunge olungenamkhawulo lwejiyomethri, kumele siqinisekise ukuthi isilinganiso \( |r| < 1 \). Ngakho-ke, isamba sochungechunge singabalwa kusetshenziswa ifomula elula neqondile. Kusukela ezinkingeni zesibonelo ezingenhla, singabona ukuthi le ndlela yenza kube lula kakhulu ukuxazulula izinkinga ezihilela uchungechunge olungenamkhawulo lwejiyomethri.
