Imibuzo Eyisibonelo Exoxa Ngochungechunge Lwejiyometri
Uchungechunge lwejiyomethri luwumqondo obalulekile kwizibalo, oluvame ukuvela ezinhlotsheni ezahlukene zezinkinga, okuhlanganisa izivivinyo zesikole, izivivinyo zokungena ekolishi, ngisho nezivivinyo ezijwayelekile njenge-SAT noma i-GRE. Ukuqonda kahle uchungechunge lwejiyomethri kusisiza ukuxazulula izinkinga ngempumelelo. Lesi sihloko sizomboza izinkinga eziningana zezibonelo futhi sixoxe ngochungechunge lwejiyomethri ngokuningiliziwe.
Ukuqonda Uchungechunge Lwejiyometri
Uchungechunge lwejiyomethri luwuchungechunge lapho igama ngalinye litholakala ngokuphindaphinda igama elidlule ngenombolo eqondile ebizwa ngokuthi isilinganiso (isilinganiso esivamile, esivame ukufanekiswa uhlamvu \(r\)). Ngokuvamile, uchungechunge lwejiyomethri lungabhalwa kanje:
\[
a, ar, ar^2, ar^3, \ldots
\]
Di mana:
– \(a\) igama lokuqala
– \(r\) yisilinganiso sochungechunge
Uma \( |r| < 1 \), uchungechunge lwe-geometric olungenamkhawulo lunezakhiwo ezithakazelisayo ezihlanganayo. Kunezindlela eziningi ezisebenzayo zochungechunge lwejiyomethri emikhakheni ehlukahlukene njengefiziksi, ezomnotho, kanye nebhayoloji. Ifomula Yochungechunge Lwejiyomethri Ithemu ye-nth yochungechunge lwejiyomethri Ithemu ye-nth yochungechunge lwejiyomethri ingabalwa kusetshenziswa ifomula: \[ U_n = a \cdot r^{n-1} \] Isamba se-n Yokuqala Imigomo Yochungechunge Lwejiyomethri Isamba semigomo yokuqala \(n\) yochungechunge lwejiyomethri (Sn) singabalwa kusetshenziswa ifomula: \[ S_n = a \frac{1 - r^n}{1 - r}, \quad \text{for } r \neq 1 \] \[ S_n = na, \quad \text{for } r = 1 \] Isamba Esingenamkhawulo Sochungechunge Lwejiyomethri Uma \(|r| < 1\), uchungechunge lwejiyomethri olungenamkhawulo lunesamba: \[ S_{\infty} = \frac{a}{1 - r} \] Imibuzo Nezingxoxo Zezibonelo Okulandelayo ezinye zezibonelo zemibuzo yochungechunge lwejiyomethri kanye nezingxoxo zayo: Isibonelo Umbuzo 1: Ukubala Umbuzo Wethemu ye-nth: Uchungechunge lwejiyomethri olunikezwe ngethemu yokuqala \(a = 5\) kanye nesilinganiso \(r = 3\). Bala itemu lesi-6 yochungechunge. Ingxoxo: Ukusebenzisa ifomula yethemu le-nth: \[ U_6 = a \cdot r^{(6-1)} = 5 \cdot 3^5 = 5 \cdot 243 = 1215 \] Ngakho-ke, ithemu yesi-6 yochungechunge ingu-1215. Isibonelo Umbuzo 2: Ukubala Isamba Semigomo Yokuqala ka-n Umbuzo: Bala isamba semigomo yokuqala engu-4 yochungechunge lwejiyometri ngetemu lokuqala \(a = 2\) kanye nesilinganiso \(r = \frac{1}{2}\). Ingxoxo: Ukusebenzisa ifomula yesamba samagama okuqala angu-\(n\): \[ S_4 = a \frac{1 - r^4}{1 - r} = 2 \frac{1 - (\frac{1}{2})^4}{1 - \frac{1}{2}} = 2 \frac{1 - \frac{1}{16}}{\frac{1}{2}} = 2 \frac{\frac{15}{16}}{\frac{1}{2}} = 2 \cdot \frac{15}{8} = 2 \cdot \frac{15}{8} = \frac{30}{8} = 3.75 \] Ngakho-ke, isamba samagama okuqala angu-4 ochungechungeni singu-3.75. Isibonelo Umbuzo 3: Isamba Sochungechunge Lwejiyomethri Olungenamkhawulo Umbuzo: Bala isamba sochungechunge olungenamkhawulo lapho \(a = 7\) kanye \(r = \frac{1}{3}\). Ingxoxo: Ukusebenzisa ifomula yesamba sochungechunge olungenamkhawulo: \[ S_{\infty} = \frac{a}{1 - r} = \frac{7}{1 - \frac{1}{3}} = \frac{7}{\frac{2}{3}} = 7 \cdot \frac{3}{2} = \frac{21}{2} = 10.5 \] Ngakho-ke, isamba sochungechunge olungenamkhawulo singu-10.5. Isibonelo Umbuzo 4: Ukunquma Imigomo Nezilinganiso Zombuzo Wochungechunge: Isamba samagama amathathu okuqala ochungechunge lwejiyometri singama-21, kanti isamba samagama esi-2 nelesi-3 singama-18. Nquma ithemu lokuqala kanye nesilinganiso salo. Ingxoxo: Ake sithi itemu lokuqala lingu-\(a\) kanti isilinganiso singu-\(r\). Kusukela kulwazi lwenkinga, singabhala lezi zibalo ezimbili ezilandelayo: \[ a + ar + ar^2 = 21 \quad \text{(1)} \] \[ ar + ar^2 = 18 \quad \text{(2)} \] Kusukela ku-equation (2), singaveza \(a\) ngokwe-\(r\): \[ a(r + r^2) = 18 \kusho a = \frac{18}{r(1 + r)} \] Okulandelayo, faka \(a\) ku-equation (1): \[ \frac{18(1)}{r(1 + r)} + \frac{18r}{r(1 + r)} + \frac{18r^2}{r(1 + r)} = 21 \] \[ \frac{18}{1 + r} + \frac{18r}{1 + r} + \frac{18r^2}{1 + r} = 21 \] \[ \frac{18 (1 + r + r^2)}{1 + r} = 21 \] \[ \frac{18 \cdot 3}{1 + r} = 21 \] \[ \frac{54}{1 + r} = 21 \] \[ 54 = 21(1 + r) \] \[ 54 = 21 + 21r \] \[ 33 = 21r \] \[ r = \frac{33}{21} = \frac{11}{7} \] Njengoba inani lika-\(r\) laziwa, libuyisele kunani lika-\(a\): \[ a = \frac{18}{r(1 + r)} = \frac{18}{\frac{11}{7} (1 + \frac{11}{7})} = \frac{18}{\frac{11}{7} \cdot \frac{18}{7}} = \frac{18 \cdot 7}{11 \cdot 18} = \frac{7}{11} \] Ngakho-ke, igama lokuqala \(a\) lingu-\(\frac{7}{11}\) kanti isilinganiso singu-\(\frac{11}{7}\). Isiphetho Uchungechunge lwe-geometric luwumqondo wezibalo osetshenziswa kabanzi ezinhlotsheni ezahlukene zokusebenza. Ukuqonda amafomula ayisisekelo njengethemu le-nth, isamba samagama okuqala ka-n, kanye nesamba sochungechunge lwejiyometri olungenamkhawulo kubalulekile ekuxazululeni izinkinga ezahlukahlukene zezibalo ezihlobene. Ngokuzijwayeza izinkinga zezibonelo ezahlukahlukene njengoba kuxoxwe ngazo kulesi sihloko, singathuthukisa ikhono lethu lokuqonda nokusebenzisa uchungechunge lwejometri kangcono.
