Nhevedzano yeJomethri: Pfungwa, Hunhu, uye Mashandisirwo
Pendauluan
Masvomhu, nekunaka kwawo kwese uye kuoma kwawo, anowanzo kupa pfungwa dzinonakidza dzine mashandisirwo anoshanda muhupenyu chaihwo. Imwe pfungwa yakadaro inoita basa rakakosha mumasvomhu uye mashandisirwo ayo igeometric series. Geometric series inopa nzira yekunzwisisa nekuongorora zviitiko zvinokura zvakanyanya kana series inoratidza mapatani chaiwo ekupetwa kaviri. Chinyorwa chino chichatsanangura zvakadzama pfungwa, hunhu, uye mashandisirwo egeometric series.
Tsanangudzo yeGeometric Series
Chitsauko che geometric inhamba dzinotevedzana umo izwi rega rega rinowanikwa nekuwedzera izwi rekare nenhamba yakatarwa inonzi ratio. Semuenzaniso, kana \( a \) iri izwi rekutanga rejiyometri uye \( r \) iri chiyero (multiplicative constant), saka chitsauko che geometric chinogona kunyorwa seizvi:
\[ a, ar, ar^2, ar^3, \ldots \]
Apo izwi rega rega rinowanikwa nekuwedzera izwi rekare nechiyero \( r \). Saka, izwi rekuti nth remutsara wejometri rinogona kuratidzwa seizvi:
\[ a_n = a \cdot r^{n-1} \]
Semuenzaniso, nhevedzano \( 2, 6, 18, 54, \ldots \) inhevedzano yejometri ine \( a = 2 \) uye \( r = 3 \) nekuti izwi rega rega rinowanikwa nekuwedzera izwi rekare ne3.
Hunhu hweGeometric Series
1. Kuwanda Kwenguva Dzose (Ratio): Hunhu hwese hwejeometric series ndeyekuti mazwi maviri akatevedzana ane constant ratio. Ichi ndicho chinhu chikuru chinosiyanisa jeometric series kana ichienzaniswa nemamwe marudzi e series kana sequences.
2. Exponential Equation: Izwi rekuti nth rechikamu che geometric rinogona kuratidzwa ne exponential equation \( a_n = a \cdot r^{n-1} \), apo \( n \) ndiyo nzvimbo yezwi iri muchikamu.
3. Huwandu hweMashoko eNhevedzano yeGeometric: Huwandu hwemashoko ekutanga \(n\) enhevedzano yegeometric hunogona kuverengerwa uchishandisa fomura:
\[ S_n = a \left( \frac{1 – r^n}{1 – r} \right) \]
ye \( r \neq 1 \). Kana \( r = 1 \), ipapo nhevedzano inova nhevedzano isingachinji uye huwandu hwayo ndi \( S_n = n \cdot a \).
4. Infinite Geometric Series: Kune infinite geometric series, huwandu hwese hwese hunopiwa na:
\[ S_{\infty} = \frac{a}{1 – r} \]
chero bedzi \( |r| < 1 \). Izvi zvinodaro nekuti nhevedzano iyi ichabatana (inosvika pamutengo wakati) kana chiyero chakazara chiri pasi pa1. Mienzaniso neMifananidzo Ngatitarisei mimwe mienzaniso kuti tijekese pfungwa yenhevedzano yejiyometri: 1. Muenzaniso weNhevedzano yeFinite Geometric: Ngatitii tine nhevedzano \( 3, 12, 48, 192, \ldots \), zvino zvinogona kuonekwa kuti: \[ a = 3 \] \[ r = 4 \] Kuti tiverenge huwandu hwemashoko mashanu ekutanga, tinogona kushandisa fomura yehuwandu hwemashoko: \[ S_5 = 3 \left( \frac{1 - 4^5}{1 - 4} \right) = 3 \left( \frac{1 - 1024}{-3} \right) = 3 \times \left( \frac{-1023}{-3} \right) = 3 \times 341 = 1023 \]
