Tambayoyi Misali Game da Jerin Geometric
Jerin Geometric muhimmin ra'ayi ne a fannin lissafi, wanda galibi ke bayyana a cikin nau'ikan matsaloli daban-daban, gami da jarrabawar makaranta, jarrabawar shiga kwaleji, har ma da gwaje-gwajen da aka daidaita kamar SAT ko GRE. Cikakken fahimtar jerin geometric yana taimaka mana mu magance matsaloli yadda ya kamata. Wannan labarin zai rufe misalai da yawa na matsaloli kuma ya tattauna jerin geometric dalla-dalla.
Fahimtar Jerin Lissafi
Jerin siffofi na lissafi jeri ne wanda ake samun kowace kalma ta hanyar ninka kalmar da ta gabata da wani adadi mai ƙayyadadden lamba da ake kira rabo (rabo na gama gari, wanda yawanci ake nuna shi da harafin \(r\)). Gabaɗaya, ana iya rubuta jerin siffofi kamar haka:
\[
a, ar, ar^2, ar^3, \ldos
\]
Ina:
– \(a\) shine kalma ta farko
– \(r\) shine rabon jerin
Idan \( |r| < 1 \), jerin siffofi marasa iyaka suna da siffa mai ban sha'awa ta haɗuwa. Akwai aikace-aikace da yawa na jerin siffofi a fannoni daban-daban kamar kimiyyar lissafi, tattalin arziki, da ilmin halitta.
Tsarin Jerin Geometric Kalma ta n na jerin geometric Kalma ta n na jerin geometric Ana iya ƙididdige kalma ta n na jerin geometric ta amfani da dabarar: \[ U_n = a \cdot r^{n-1} \] Jimlar Kalmomin N na Farko na Jerin Geometric Ana iya ƙididdige jimlar kalmomin \(n\) na farko na jerin geometric (Sn) ta amfani da dabarar: \[ S_n = a \frac{1 - r^n}{1 - r}, \quad \text{for } r \neq 1 \] \[ S_n = na, \quad \text{for } r = 1 \] Jimlar Infinite na Jerin Geometric If \(|r| < 1\), jerin geometric mara iyaka yana da jimlar: \[ S_{\infty} = \frac{a}{1 - r} \] Tambayoyi da Tattaunawa Misali Ga wasu misalan tambayoyin jerin geometric tare da tattaunawarsu: Misali Tambaya ta 1: Lissafin Jadawali na n na Tambaya: An ba da jerin geometric tare da kalma ta farko \(a = 5\) da rabo na gama gari \(r = 3\). Lissafa kalma ta 6 ta jerin. Magani: Amfani da dabarar kalma ta n: \[U_6 = a \cdot r^{(6-1)} = 5 \cdot 3^5 = 5 \cdot 243 = 1215 \] Don haka, kalma ta 6 ta jerin ita ce 1215. Misali Tambaya ta 2: Lissafin Jimlar Kalmomin n na Farko Tambaya: A ƙididdige jimlar kalmomin farko guda 4 na jerin lissafi tare da kalmar farko ta \(a = 2\) da rabon \(r = \frac{1}{2}\). Tattaunawa: Amfani da dabarar jimlar kalmomin farko na \(n\): \[S_4 = a \frac{1 - r^4}{1 - r} = 2 \frac{1 - (\frac{1}{2})^4}{1 - \frac{1}{2}} = 2 \frac{1 - \frac{1}{16}}{\frac{1}{2}} = 2 \frac{\frac{15}{16}}{\frac{1}{2}} = 2 \cdot \frac{15}{8} = 2 \cdot \frac{15}{8} = \frac{30}{8} = 3.75 \] Don haka, jimlar kalmomin farko guda 4 na jerin shine 3.75. Misali na 3: Jimlar Tambayar Jerin Jerin Geometric Mara Iyaka: Lissafa jimlar jerin jeri mara iyaka inda \(a = 7\) da \(r = \frac{1}{3}\). Magani: Amfani da dabarar jimlar jerin jeri mara iyaka: \[S_{\infty} = \frac{a}{1 - r} = \frac{7}{1 - \frac{1}{3}} = \frac{7}{\frac{2}{3}} = 7 \cdot \frac{3}{2} = \frac{21}{2} = 10.5 \] Don haka, jimlar jerin jeri mara iyaka shine 10.5. Misali na 4: Ƙayyade Sharuɗɗa da Rabon Tambayar Jeri: Jimlar sharuɗɗan farko 3 na jerin geometric shine 21, kuma jimlar sharuɗɗan 2 da 3 shine 18. Ƙayyade kalma ta farko da rabonta. Tattaunawa: A ce kalma ta farko ita ce \(a\) kuma rabon shine \(r\). Daga bayanan matsala, za mu iya rubuta waɗannan daidaito guda biyu: \[a + ar + ar^2 = 21 \quad \text{(1)} \] \[ar + ar^2 = 18 \quad \text{(2)} \] Daga lissafi (2), za mu iya bayyana \(a\) dangane da \(r\): \[a(r + r^2) = 18 \na nuna a = \frac{18}{r(1 + r)} \] Na gaba, maye gurbin \(a\) zuwa lissafi (1): \[ \frac{18(1)}{r(1 + r)} + \frac{18r}{r(1 + r)} + \frac{18r^2}{r(1 + r)} = 21 \] \[ \frac{18}{1 + r} + \frac{18r}{1 + r} + \frac{18r^2}{1 + r} = 21 \] \[ \frac{18 (1 + r + r^2)}{1 + r} = 21 \] \[ \frac{18 \cdot 3}{1 + r} = 21 \] \[ \frac{54}{1 + r} = 21 \] \[ 54 = 21(1 + r) \] \[ 54 = 21(1 + r) \] \[ 54 = 21 + 21r \] \[ 33 = 21r \] \[ r = \frac{33}{21} = \frac{11}{7} \] Da zarar an san ƙimar \(r\), a mayar da ita zuwa ƙimar \(a\): \[ a = \frac{18}{r(1 + r)} = \frac{18}{\frac{11}{7} (1 + \frac{11}{7})} = \frac{18}{\frac{11}{7} \cdot \frac{18}{7}} = \frac{18 \cdot 7}{11 \cdot 18} = \frac{7}{11} \] Don haka, kalmar farko \(a\) shine \(\frac{7}{11}\) kuma rabon da aka saba dashi shine \(\frac{11}{7}\). Kammalawa Jerin Geometric yana ɗaya daga cikin ra'ayoyin lissafi da ake amfani da su sosai a aikace-aikace daban-daban. Fahimtar dabarun asali kamar kalmar n, jimlar kalmomin n na farko, da jimlar jerin geometric mara iyaka yana da matuƙar mahimmanci don magance matsalolin lissafi daban-daban masu alaƙa. Ta hanyar yin amfani da misalai daban-daban kamar yadda aka tattauna a cikin wannan labarin, za mu iya inganta ƙwarewarmu ta fahimta da amfani da jerin geometric mafi kyau.