Mienzaniso yeMibvunzo Inokurukura nezveGeometric Series
Kutevedzana kwejiyometri ipfungwa huru mumasvomhu inowanzo dzidziswa kuchikoro chesekondari. Kutevedzana uku kunosanganisira nhamba, imwe neimwe iri chibereko chenhamba yapfuura ne "constant" inozivikanwa se "ratio." Chinyorwa chino chichataura nezvematambudziko akati wandei emuenzaniso uye hurukuro dzezvikamu zvejiyometri, tichitarisira kubatsira vaverengi kunzwisisa pfungwa iyi zviri nani.
Kunzwisisa Nhevedzano yeGeometric
Kutevedzana kwejiyometri (geometric sequence) kutevedzana kwenhamba kunoumbwa nekupeta nhamba yekutanga (a) nechiyero chakagadzika (r). Kazhinji, chimiro chakajairika che kutevedzana kwejiyometri ndeichi:
\[ a, ar, ar^2, ar^3, \ldots, ar^{n-1} \]
Pano:
– “a” izwi rekutanga remutsara uyu.
– “r” zvinoreva chiyero chezwi rimwe chete kune izwi rapfuura.
– “n” izwi rechina (nth) munhevedzano.
Mibvunzo yemuenzaniso nekukurukurirana
Ngatikurukurei mimwe mienzaniso yezvinetso kuti tinzwisise zvakawanda nezve geometric sequences.
Muenzaniso Mubvunzo 1
Mubvunzo:
Zvichienderana nehurongwa hwe geometric, izwi rekutanga (a) riri 3 uye chiyero (r) chiri 2. Sarudza:
1. Chikamu chegumi chemutsara.
2. Huwandu hwemashoko matanhatu ekutanga emutsara.
Kukurukurirana:
1. Temu yechishanu (U5) inogona kuverengerwa uchishandisa fomura yetemu yechitanhatu yenhevedzano yejometri, inoti:
\[ U_n = a \cdot r^{n-1} \]
Kuisa a = 3, r = 2, uye n = 5 mufomura:
\[ U_5 = 3 \cdot 2^{5-1} \]
\[ U_5 = 3 \cdot 2^4 \]
\[ U_5 = 3 \cdot 16 \]
\[ U_5 = 48 \]
Saka, chikamu chechishanu chemutsara uyu i48.
2. Huwandu hwemashoko matanhatu ekutanga (S6) ejeometrical sequence hunogona kuverengerwa uchishandisa fomura yehuwandu hwemashoko ekutanga n, anoti:
\[ S_n = a \left( \frac{r^n – 1}{r – 1} \right) \]
Kuisa a = 3, r = 2, uye n = 6 mufomura:
\[ S_6 = 3 \kuruboshwe( \frac{2^6 – 1}{2 – 1} \kurudyi) \]
\[ S_6 = 3 \kuruboshwe( \frac{64 – 1}{1} \kurudyi) \]
\[ S_6 = 3 \kuruboshwe( 63 \kurudyi) \]
\[ S_6 = 189 \]
Saka, huwandu hwemashoko matanhatu ekutanga emutsara uyu i189.
Muenzaniso Mubvunzo 2
Mubvunzo:
Kutevedzana kwejometri kune chikamu chechitatu che27 uye chikamu chechishanu che243. Sarudza kukosha kwechikamu chekutanga (a) uye chiyero (r).
Kukurukurirana:
Zvichipiwa U3 = 27 uye U5 = 243. Kushandisa fomura yenth term ye geometric sequence:
\[ U_n = a \cdot r^{n-1} \]
Kune U3:
\[ U_3 = a \cdot r^2 \]
\[ 27 = a \cdot r^2 \] \[ (1) \]
Kune U5:
\[ U_5 = a \cdot r^4 \]
\[ 243 = a \cdot r^4 \] \[ (2) \]
Kuenzanisa maequation (1) na (2) kubvisa a:
\[ \frac{U_5}{U_3} = \frac{a \cdot r^4}{a \cdot r^2} \]
\[ \frac{243}{27} = r^2 \]
\[ 9 = r^2 \]
\[r = 3 \chinyorwa{ kana } r = -3 \]
Isa kukosha kwe r mu equation (1):
Kana \(r = 3 \):
\[ 27 = a \cdot 3^2 \]
\[ 27 = a \cdot 9 \]
\[ a = 3 \]
Kana \(r = -3 \):
\[ 27 = a \cdot (-3)^2 \]
\[ 27 = a \cdot 9 \]
\[ a = 3 \]
Saka, izwi rekutanga (a) ndi3, uye chiyero (r) chinogona kuva 3 kana -3.
Muenzaniso Mubvunzo 3
Mubvunzo:
Tsvaga huwandu husingaperi hwema geometric series anotevera kana izwi rekutanga (a) riri 8 uye chiyero (r) chiri 1/2.
Kukurukurirana:
Huwandu husingaperi hwejeometri series hunogona kuverengerwa uchishandisa fomura:
\[ S_{\infty} = \frac{a}{1 – r} \]
Kuisa a = 8 uye r = 1/2 mufomura:
\[ S_{\infty} = \frac{8}{1 – \frac{1}{2}} \]
\[ S_{\infty} = \frac{8}{\frac{1}{2}} \]
\[ S_{\infty} = 8 \kawa 2 \]
\[ S_{\infty} = 16 \]
Saka, huwandu husingaperi hwejeometriyamu i16.
Muenzaniso Mubvunzo 4
Mubvunzo:
Kutevedzana kwejometri kune chikamu chechipiri chegumi nembiri uye chikamu chechina chegumi nesere chegumi nesere. Sarudza chiyero uye chikamu chekutanga chechikamu.
Kukurukurirana:
Zvapiwa \( U_2 = 12 \) uye \( U_4 = 108 \). Kushandisa fomura yetemu yechitanhatu yejeometri:
Kune \( U_2 \):
\[ U_2 = a \cdot r \]
\[ 12 = a \cdot r \] \[ (1) \]
Kune \( U_4 \):
\[ U_4 = a \cdot r^3 \]
\[ 108 = a \cdot r^3 \] \[ (2) \]
Kuenzanisa maequation (1) na (2) kubvisa a:
\[ \frac{U_4}{U_2} = \frac{a \cdot r^3}{a \cdot r} \]
\[ \frac{108}{12} = r^2 \]
\[ 9 = r^2 \]
\[r = 3 \chinyorwa{ kana } r = -3 \]
Isa kukosha kwe r mu equation (1):
Kana \(r = 3 \):
\[ 12 = a \cdot 3 \]
\[ a = 4 \]
Kana \(r = -3 \):
\[ 12 = a \cdot (-3) \]
\[ a = -4 \]
Saka, izwi rekutanga (a) rinogona kuva 4 kana -4, uye chiyero (r) chinogona kuva 3 kana -3.
Mhedziso
Kutevedzana kwejiyometri ipfungwa inokosha yemasvomhu inowanzoshandiswa muminda yakasiyana-siyana. Nekunzwisisa zvekutanga uye kudzidzira hunyanzvi hwekugadzirisa matambudziko, tinotarisira kuti tinogona kuva nehunyanzvi mukunzwisisa nekushandisa pfungwa iyi. Chinyorwa chino chine mienzaniso yakawanda yezvinetso nehurukuro dzekubatsira vaverengi kudzidza nekunzwisisa kutevedzana kwejiyometri zvakadzama. Tinovimba izvi zvinobatsira!
