Tambayoyi Misali Game da Jerin Geometric Mara Iyaka
Jerin siffofi marasa iyaka jerin kalmomi ne da suka ƙunshi kalmomi marasa iyaka a cikin ci gaban siffofi. Waɗannan jerin suna da takamaiman buƙatu don a iya ƙididdige jimlarsu (ko iyakarsu). A cikin wannan labarin, za mu tattauna ainihin manufar jerin siffofi marasa iyaka, buƙatun gina su, da misalai da yawa na matsaloli da mafita.
Ma'anar Asali na Jerin Geometric Mara Iyaka
Ainihin, jerin lambobi na lissafi jerin lambobi ne wanda ake samun kowace kalma bayan ta farko ta hanyar ninka kalmar da ta gabata da wani tsari da ake kira rabo na gama gari (r). A ce muna da jerin lambobi na lissafi:
\[a, ar, ar^2, ar^3, ar^4, \ldots \]
Ga jerin siffofi marasa iyaka, muna la'akari da jimlar dukkan kalmomi a cikin jerin. Jimlar wannan jerin an bayyana shi kamar haka:
\[ S = a + ar + ar^2 + ar^3 + ar^4 + \ldots \]
Jimlar jerin siffofi marasa iyaka suna haɗuwa (yana da takamaiman jimla) idan kuma kawai idan rabon \( |r| < 1 \). Idan \( |r| \geq 1 \), to jerin sun bambanta kuma ba su da takamaiman jimla (suna zuwa rashin iyaka).
Idan \( |r| < 1 \), jimlar S na jerin lissafi mara iyaka za a iya bayyana shi kamar haka: \[ S = \frac{a}{1-r} \] inda: - \( S \) shine jimlar jerin, - \( a \) shine kalma ta farko, - \( r \) shine rabo. Misali Tambayoyi da Tattaunawa Misali Tambaya ta 1 Tambaya: Nemo jimlar jerin lissafi mara iyaka don jerin masu zuwa: \[ 3 + 1.5 + 0.75 + 0.375 + \ldots \] Tattaunawa: Bari mu gano muhimman abubuwan da ke cikin jerin: Kalma ta farko \( a = 3 \) Ana iya samun rabon \( r \) ta hanyar raba kalma ta biyu da kalma ta farko, wato: \[ r = \frac{1.5}{3} = 0.5 \] Tunda \( |r| = 0.5 < 1 \), wannan jerin ya haɗu kuma za mu iya ƙididdige jimlar jerin marasa iyaka. Yi amfani da dabarar don jimlar jerin lissafi mara iyaka: \[ S = \frac{a}{1-r} \] \[ S = \frac{3}{1-0.5} \] \[ S = \frac{3}{0.5} \] \[ S = 6 \] Don haka, jimlar jerin lissafi mara iyaka shine 6. Misali Tambaya ta 2 Tambaya: Nemo jimlar jerin lissafi mara iyaka tare da kalma ta farko 8 da rabo \( r = -\frac{1}{3} \). Tattaunawa: Kalma ta farko \( a = 8 \) Ratio \( r = -\frac{1}{3} \) Tunda \( |r| = \frac{1}{3} < 1 \), wannan jerin ya haɗu kuma za mu iya ƙididdige jimlar jerin marasa iyaka. Yi amfani da dabarar don jimlar jerin jeri na geometric mara iyaka: \[ S = \frac{a}{1-r} \] \[ S = \frac{8}{1 - \left(-\frac{1}{3}\right)} \] \[ S = \frac{8}{1 + \frac{1}{3}} \] \[ S = \frac{8}{\frac{4}{3}} \] \[ S = 8 \times \frac{3}{4} \] \[ S = 6 \] Don haka, jimlar jerin jeri na geometric mara iyaka shine 6. Misali Tambaya ta 3 Tambaya: Shin jerin da ke ƙasa suna da jimlar marasa iyaka? Idan haka ne, nemo jimlar. \[ 5 + 2.5 + 1.25 + 0.625 + \ldots \] Tattaunawa: Kalma ta farko \( a = 5 \) Ana iya samun rabon \( r \) ta hanyar raba kalma ta biyu da kalma ta farko, wato: \[ r = \frac{2.5}{5} = 0.5 \] Tunda \( |r| = 0.5 < 1 \), wannan jerin ya haɗu kuma za mu iya ƙididdige jimlar jerin marasa iyaka. Yi amfani da dabarar don jimlar jerin lissafi marasa iyaka: \[ S = \frac{a}{1-r} \] \[ S = \frac{5}{1-0.5} \] \[ S = \frac{5}{0.5} \] \[ S = 10 \] Don haka, jimlar jerin lissafi marasa iyaka shine 10. Misali Tambaya ta 4 Tambaya: Tantance ko jerin da ke ƙasa sun haɗu ko sun bambanta: \[ 4 - 6 + 9 - 13.5 + \ldots \] Tattaunawa: Kalma ta farko \( a = 4 \) Ana iya samun rabon \( r \) ta hanyar raba kalma ta biyu da kalma ta farko, wato: \[ r = \frac{-6}{4} = -1.5 \] Tunda \( |r| = 1.5 > 1 \), wannan jerin ya bambanta kuma ba shi da wani takamaiman jimilla.Don haka, jerin ya bambanta.
Misali Tambaya ta 5
Tambaya: A ce kana da jerin marasa iyaka masu zuwa:
\[ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots \]
Ƙayyade jimlar jerin.
Tattaunawa:
Kalma ta farko \( a = \frac{1}{2} \)
Za a iya samun rabon \( r \) ta hanyar raba kalma ta biyu da kalma ta farko, wato:
\[ r = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2} \]
Tunda \( |r| = \frac{1}{2} < 1 \), wannan jerin ya haɗu kuma za mu iya ƙididdige jimlar jerin marasa iyaka. Yi amfani da dabarar don jimlar jerin jeri na geometric mara iyaka: \[ S = \frac{a}{1-r} \] \[ S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} \] \[ S = \frac{\frac{1}{2}}{\frac{1}{2}} \] \[ S = 1 \] Don haka, jimlar jerin jeri na geometric mara iyaka shine 1. Kammalawa Jeri na geometric mara iyaka muhimmin ra'ayi ne na lissafi wanda ke da fa'idodi da yawa a fannoni daban-daban. Don tantance jimlar jerin jeri na geometric mara iyaka, dole ne mu tabbatar da cewa rabon \( |r| < 1 \). Don haka, ana iya ƙididdige jimlar jerin ta amfani da dabara mai sauƙi da sauƙi. Daga misalan matsalolin da ke sama, za mu iya ganin cewa wannan dabarar tana sa ya zama mai sauƙin magance matsalolin da suka shafi jerin jeri na geometric mara iyaka.
