Mienzaniso yeMibvunzo Inokurukura nezveGeometric Series
Ma "geometric series" ipfungwa yakakosha mumasvomhu, inowanzoonekwa mumhando dzakasiyana dzematambudziko, kusanganisira bvunzo dzechikoro, bvunzo dzekupinda kukoreji, uye kunyangwe bvunzo dzakajairwa dzakadai seSAT kana GRE. Kunzwisisa kwakakwana ma "geometric series" kunotibatsira kugadzirisa matambudziko zvinobudirira. Chinyorwa chino chichataura nezvemienzaniso yakati wandei yematambudziko uye chichakurukura ma "geometric series" zvakadzama.
Kunzwisisa Nhevedzano yeGeometric
Chitsauko che geometric inyaya iyo izwi rega rega rinowanikwa nekuwedzera izwi rekare nenhamba yakatarwa inonzi ratio (common ratio, inowanzo miririrwa nebhii \(r\)). Kazhinji, chitsauko che geometric chinogona kunyorwa seizvi:
\[
a, ar, ar^2, ar^3, \ldots
\]
Di mana:
– \(a\) izwi rekutanga
– \(r\) ndicho chiyero chenhevedzano
Kana \( |r| < 1 \), mutsetse we geometric usingaperi une hunhu hunonakidza hwekubatana. Kune mashandisirwo akawanda anoshanda emutsetse we geometric muzvikamu zvakasiyana-siyana zvakaita sefizikisi, economics, uye biology.
Fomura reGeometric Series Izwi rekuti nth remutsara wejometri Izwi rekuti nth remutsara wejometri rinogona kuverengerwa uchishandisa fomura: \[ U_n = a \cdot r^{n-1} \] Huwandu hweFirst n Terms of a Geometric Series Huwandu hwe \(n\) mazwi ekutanga emutsara wejometri (Sn) hunogona kuverengerwa uchishandisa fomura: \[ S_n = a \frac{1 - r^n}{1 - r}, \quad \text{for } r \neq 1 \] \[ S_n = na, \quad \text{for } r = 1 \] Huwandu Husingaperi hweMutsetse weJometri Kana \(|r| < 1\), mutsara wejometri usingaperi une huwandu: \[ S_{\infty} = \frac{a}{1 - r} \] Muenzaniso weMibvunzo neKukurukurirana Izvi zvinotevera mimwe mienzaniso yemibvunzo yemutsara wejometri pamwe chete nehurukuro dzavo: Muenzaniso Mubvunzo 1: Kuverenga Mubvunzo weTemu yenth: Yakapihwa mutsara wejometri netemu yekutanga \(a = 5\) uye common ratio \(r = 3\). Verenga term yechitanhatu yenhevedzano. Mhinduro: Uchishandisa fomura yenth term: \[ U_6 = a \cdot r^{(6-1)} = 5 \cdot 3^5 = 5 \cdot 243 = 1215 \] Saka, term yechitanhatu yenhevedzano i1215. Muenzaniso Mubvunzo 2: Kuverenga Huwandu hweMashoko ekutanga n Mubvunzo: Verenga huwandu hwemashoko mana ekutanga emutsara wejometri netemu yekutanga \(a = 2\) uye chiyero \(r = \frac{1}{2}\). Kukurukurirana: Kushandisa fomura yehuwandu hwemashoko ekutanga \(n\): \[ S_4 = a \frac{1 - r^4}{1 - r} = 2 \frac{1 - (\frac{1}{2})^4}{1 - \frac{1}{2}} = 2 \frac{1 - \frac{1}{16}}{\frac{1}{2}} = 2 \frac{\frac{15}{16}}{\frac{1}{2}} = 2 \cdot \frac{15}{8} = 2 \cdot \frac{15}{8} = \frac{30}{8} = 3.75 \] Saka, huwandu hwemashoko mana ekutanga emutsara i3.75. Muenzaniso 3: Huwandu hweMutsetse weGeometric Usingaperi Mubvunzo: Verenga huwandu hwemutsetse usingaperi apo \(a = 7\) uye \(r = \frac{1}{3}\). Mhinduro: Kushandisa fomura yehuwandu hwemutsetse usingaperi: \[ S_{\infty} = \frac{a}{1 - r} = \frac{7}{1 - \frac{1}{3}} = \frac{7}{\frac{2}{3}} = 7 \cdot \frac{3}{2} = \frac{21}{2} = 10.5 \] Saka, huwandu hwemutsetse usingaperi i10.5. Muenzaniso 4: Kusarudza Mitemo neChiyero cheMutsetse Mubvunzo: Huwandu hwemashoko matatu ekutanga emutsetse wegeometric i21, uye huwandu hwemashoko echipiri nerechitatu i18. Sarudza izwi rekutanga nechiyero charo. Kukurukurirana: Ngatitii izwi rekutanga i \(a\) uye chiyero i \(r\). Kubva paruzivo rwedambudziko, tinogona kunyora ma equation maviri anotevera: \[ a + ar + ar^2 = 21 \quad \text{(1)} \] \[ ar + ar^2 = 18 \quad \text{(2)} \] Kubva pa equation (2), tinogona kuratidza \(a\) maererano ne \(r\): \[ a(r + r^2) = 18 \zvinoreva a = \frac{18}{r(1 + r)} \] Tevere, tsiva \(a\) mu equation (1): \[ \frac{18(1)}{r(1 + r)} + \frac{18r}{r(1 + r)} + \frac{18r^2}{r(1 + r)} = 21 \] \[ \frac{18}{1 + r} + \frac{18r}{1 + r} + \frac{18r^2}{1 + r} = 21 \] \[ \frac{18 (1 + r + r^2)}{1 + r} = 21 \] \[ \frac{18 \cdot 3}{1 + r} = 21 \] \[ \frac{54}{1 + r} = 21 \] \[ 54 = 21(1 + r) \] \[ 54 = 21 + 21r \] \[ 33 = 21r \] \[ r = \frac{33}{21} = \frac{11}{7} \] Kana kukosha kwe \(r\) kwazivikanwa, dzorera kukosha kwe \(a\): \[ a = \frac{18}{r(1 + r)} = \frac{18}{\frac{11}{7} (1 + \frac{11}{7})} = \frac{18}{\frac{11}{7} \cdot \frac{18}{7}} = \frac{18 \cdot 7}{11 \cdot 18} = \frac{7}{11} \] Saka, izwi rekutanga \(a\) ndi \(\frac{7}{11}\) uye chiyero chakajairika ndi \(\frac{11}{7}\). Mhedziso Geometric series ndeimwe yepfungwa dzemasvomhu dzinoshandiswa zvakanyanya mumashandisirwo akasiyana-siyana. Kunzwisisa mafomula ekutanga akadai senth term, huwandu hwemashoko ekutanga n, uye huwandu hweinfinite geometric series kwakakosha zvikuru kugadzirisa matambudziko akasiyana-siyana ane chekuita nemasvomhu. Nekudzidzira mienzaniso yakasiyana-siyana sezvakakurukurwa muchinyorwa chino, tinogona kunatsiridza kugona kwedu kunzwisisa nekushandisa geometric series zviri nani.