Fa'ata'ita'iga o Fesili e Talanoaina ai Fa'asologa Fa'ata'ita'i
O fa'asologa fa'ata'ita'i o se manatu taua tele i le matematika, e masani ona aliali mai i ituaiga eseese o fa'afitauli, e aofia ai su'ega a'oga, su'ega ulufale i kolisi, ma e o'o lava i su'ega fa'atulagaina e pei o le SAT po'o le GRE. O se malamalamaga mae'ae'a i fa'asologa fa'ata'ita'i e fesoasoani ia i tatou e fo'ia fa'afitauli ma le lelei. O lenei tusiga o le a aofia ai ni fa'ata'ita'iga o fa'afitauli ma talanoaina auiliili ai fa'asologa fa'ata'ita'i.
Malamalama i le Faasologa Fa'ata'ita'i
O se faasologa fa'ata'ita'i o se faasologa lea e maua ai vaega ta'itasi e ala i le fa'ateleina o le vaega muamua i se numera tumau e ta'ua o le fua fa'atatau (fua fa'atatau masani, e masani ona fa'ailogaina e le mata'itusi \(r\)). I se tulaga lautele, e mafai ona tusia se faasologa fa'ata'ita'i e pei o:
\[
a, ar, ar^2, ar^3, \ldots
\]
O fea:
– o le \(a\) o le ulua'i vaega lea
– \(r\) o le fua faatatau o le faasologa
Afai \( |r| < 1 \), o fa'asologa fa'ata'atitia e le gata e iai le uiga manaia o le fa'atasiga. E tele fa'aoga aoga o fa'asologa fa'ata'atitia i vaega eseese e pei o le fisiki, tamaoaiga, ma le paiolo.
Fua Fa'asologa Fa'ata'ita'i O le vaega lona 9 o se fa'asologa fa'ata'ita'i E mafai ona fuafuaina le vaega lona 9 o se fa'asologa fa'ata'ita'i e fa'aaoga ai le fua fa'atatau: \[ U_n = a \cdot r^{n-1} \] Aofa'i o le n Muamua o le Fa'asologa Fa'ata'ita'i O le aofa'i o le \(n\) muamua o le fa'asologa fa'ata'ita'i (Sn) e mafai ona fuafuaina e fa'aaoga ai le fua fa'atatau: \[ S_n = a \frac{1 - r^n}{1 - r}, \quad \text{for } r \neq 1 \] \[ S_n = na, \quad \text{for } r = 1 \] Aofa'i e le gata o se Fa'asologa Fa'ata'ita'i Afai \(|r| < 1\), o se fa'asologa fa'ata'ita'i e le gata e iai le aofa'i: \[ S_{\infty} = \frac{a}{1 - r} \] Fa'ata'ita'iga Fesili ma Talanoaga O nisi nei o fa'ata'ita'iga o fesili fa'asologa fa'ata'ita'i fa'atasi ai ma a latou talanoaga: Fa'ata'ita'iga Fesili 1: Fuafuaina o le Fesili o le Vaega lona 9: Tu'uina atu le fa'asologa fa'ata'ita'i ma le vaega muamua \(a = 5\) ma le fua fa'atatau masani \(r = 3\). Fa'atatau le vaega lona 6 o le fa'asologa. Tali: Fa'aaoga le fua fa'atatau o le vaega lona 9: \[ U_6 = a \cdot r^{(6-1)} = 5 \cdot 3^5 = 5 \cdot 243 = 1215 \] O lea la, o le vaega lona 6 o le fa'asologa o le 1215. Fa'ata'ita'iga Fesili 2: Fa'atatauina o le Aofai o Vaega Muamua n Fesili: Fuafua le aofa'i o ulua'i vaega e 4 o se fa'asologa fa'ata'atitia fa'atasi ai ma le ulua'i vaega \(a = 2\) ma le fua fa'atatau \(r = \frac{1}{2}\). Talanoaga: Fa'aaogaina le fua fa'atatau mo le aofa'i o ulua'i vaega \(n\): \[ S_4 = a \frac{1 - r^4}{1 - r} = 2 \frac{1 - (\frac{1}{2})^4}{1 - \frac{1}{2}} = 2 \frac{1 - \frac{1}{16}}{\frac{1}{2}} = 2 \frac{\frac{15}{16}}{\frac{1}{2}} = 2 \cdot \frac{15}{8} = 2 \cdot \frac{15}{8} = \frac{30}{8} = 3.75 \] O lea la, o le aofa'i o ulua'i vaega e 4 o le fa'asologa e 3.75. Faʻataʻitaʻiga 3: Fesili e uiga i le Aofai o se Faʻasologa Faʻataʻitaʻi e le Uma: Fuafua le aofaʻi o se faʻasologa e le uma lea \(a = 7\) ma \(r = \frac{1}{3}\). Tali: Faʻaaogaina le fua faʻatatau mo le aofaʻi o se faʻasologa e le uma: \[ S_{\infty} = \frac{a}{1 - r} = \frac{7}{1 - \frac{1}{3}} = \frac{7}{\frac{2}{3}} = 7 \cdot \frac{3}{2} = \frac{21}{2} = 10.5 \] O lea la, o le aofaʻi o le faʻasologa e le uma e 10.5. Faʻataʻitaʻiga 4: Fuafuaina o Faʻasologa ma le Fua Faʻatatau o se Faʻasologa Fesili e uiga i le: O le aofaʻi o uluaʻi vaega e 3 o se faʻasologa faʻataʻitaʻi e 21, ma le aofaʻi o le lona 2 ma le lona 3 o le 18. Fuafua le vaega muamua ma lona fua faʻatatau. Talanoaga: Faʻapea o le vaega muamua o le \(a\) ma o le fua faʻatatau o le \(r\). Mai faʻamatalaga o le faʻafitauli, e mafai ona tatou tusia fua faʻatatau e lua nei: \[ a + ar + ar^2 = 21 \quad \text{(1)} \] \[ ar + ar^2 = 18 \quad \text{(2)} \] Mai le fua fa'atatau (2), e mafai ona tatou fa'aalia le \(a\) i tulaga o le \(r\): \[ a(r + r^2) = 18 \implies a = \frac{18}{r(1 + r)} \] Sosoo ai, sui le \(a\) i le fua fa'atatau (1): \[ \frac{18(1)}{r(1 + r)} + \frac{18r}{r(1 + r)} + \frac{18r^2}{r(1 + r)} = 21 \] \[ \frac{18}{1 + r} + \frac{18r}{1 + r} + \frac{18r^2}{1 + r} = 21 \] \[ \frac{18 (1 + r + r^2)}{1 + r} = 21 \] \[ \frac{18 \cdot 3}{1 + r} = 21 \] \[ \frac{54}{1 + r} = 21 \] \[ 54 = 21(1 + r) \] \[ 54 = 21 + 21r \] \[ 33 = 21r \] \[ r = \frac{33}{21} = \frac{11}{7} \] Faatasi ai ma le tau o le \(r\) ua iloa, sui i le tau o le \(a\): \[ a = \frac{18}{r(1 + r)} = \frac{18}{\frac{11}{7} (1 + \frac{11}{7})} = \frac{18}{\frac{11}{7} \cdot \frac{18}{7}} = \frac{18 \cdot 7}{11 \cdot 18} = \frac{7}{11} \] O lea la, o le ulua'i vaega \(a\) o le \(\frac{7}{11}\) ma o le fua fa'atatau masani o le \(\frac{11}{7}\). Fa'ai'uga O fa'asologa fa'ata'ita'i o se tasi lea o manatu fa'amatematika e fa'aaogaina lautele i fa'aoga eseese. O le malamalama i fua fa'atatau fa'avae e pei o le vaega lona 9, le aofa'i o ulua'i vaega n, ma le aofa'i o se fa'asologa fa'ata'ita'i e le gata e matua taua tele e fo'ia ai fa'afitauli fa'amatematika eseese e feso'ota'i. I le fa'ata'ita'iina o fa'ata'ita'iga eseese e pei ona talanoaina i lenei tusiga, e mafai ona tatou fa'aleleia atili lo tatou tomai e malamalama ma fa'aaoga lelei ai fa'asologa fa'ata'ita'i.
