Mehlala ea Lipotso Tse Buisanang ka Letoto la Lijiometri
Letoto la jeometri ke mohopolo oa bohlokoa lipalo, o hlahang khafetsa mefuteng e fapaneng ea mathata, ho kenyeletsoa litlhahlobo tsa sekolo, litlhahlobo tsa ho kena kolecheng, esita le litekong tse tloaelehileng tse kang SAT kapa GRE. Kutloisiso e felletseng ea letoto la jeometri e re thusa ho rarolla mathata ka katleho. Sengoloa sena se tla akaretsa mathata a 'maloa a mehlala le ho buisana ka letoto la jeometri ka botlalo.
Ho Utloisisa Letoto la Lijiometri
Letoto la jeometri ke letoto leo ho lona lentsoe ka leng le fumanoang ka ho atisa lentsoe le fetileng ka nomoro e tsitsitseng e bitsoang karolelano (karolelano e tloaelehileng, hangata e tšoantšetsoang ke tlhaku \(r\)). Ka kakaretso, letoto la jeometri le ka ngoloa tjena:
\[
a, ar, ar^2, ar^3, \ldots
\]
Di mana:
– \(a\) ke lentsoe la pele
– \(r\) ke karolelano ea letoto
Haeba \( |r| < 1 \), letoto la jiometri le sa feleng le na le thepa e khahlisang e kopaneng. Ho na le lits'ebeliso tse ngata tse sebetsang tsa letoto la jiometri mafapheng a fapaneng a kang fisiks, moruo le baeloji. Foromo ea Letoto la Jeometrike Nako ea bo-n ea letoto la jeometrike Nako ea bo-n ea letoto la jeometrike e ka baloa ho sebelisoa foromo: \[ U_n = a \cdot r^{n-1} \] Kakaretso ea Mantsoe a Pele a n a Letoto la Jeometrike Kakaretso ea mantsoe a pele a \(n\) a letoto la jeometrike (Sn) e ka baloa ho sebelisoa foromo: \[ S_n = a \frac{1 - r^n}{1 - r}, \quad \text{for} r \neq 1 \] \[ S_n = na, \quad \text{for} r = 1 \] Kakaretso e sa Feleng ea Letoto la Jeometrike Haeba \(|r| < 1\), letoto la jeometrike le sa Feleng le na le kakaretso: \[ S_{\infty} = \frac{a}{1 - r} \] Mehlala ea Lipotso le Lipuisano Mehlala e latelang ke e meng ea lipotso tsa letoto la jeometrike hammoho le lipuisano tsa tsona: Mohlala Potso ea 1: Ho Bala Potso ea Kotara ea bo-n: Ho fanoa ka letoto la jeometrike ka kotara ea pele \(a = 5\) le karolelano \(r = 3\). Bala nako ea 6 ea letoto. Puisano: Ho sebedisa foromo ya kotara ya nth: \[ U_6 = a \cdot r^{(6-1)} = 5 \cdot 3^5 = 5 \cdot 243 = 1215 \] Kahoo, kotara ya botshelela ya letoto ke 1215. Mohlala Potso ea 2: Ho Bala Kakaretso ea Mantsoe a Pele a n Potso: Bala kakaretso ea mantsoe a 4 a pele a letoto la jiometri ka lentsoe la pele \(a = 2\) le karolelano \(r = \frac{1}{2}\). Puisano: Ho sebedisa foromo ya kakaretso ya mantswe a pele a \(n\): \[ S_4 = a \frac{1 - r^4}{1 - r} = 2 \frac{1 - (\frac{1}{2})^4}{1 - \frac{1}{2}} = 2 \frac{1 - \frac{1}{16}}{\frac{1}{2}} = 2 \frac{\frac{15}{16}}{\frac{1}{2}} = 2 \cdot \frac{15}{8} = 2 \cdot \frac{15}{8} = \frac{30}{8} = 3.75 \] Kahoo, kakaretso ya mantswe a pele a 4 a letoto ke 3.75. Mohlala Potso ea 3: Kakaretso ea Letoto la Jeometri le sa Feleng Potso: Bala kakaretso ea letoto le sa Feleng moo \(a = 7\) le \(r = \frac{1}{3}\). Puisano: Ho sebelisoa foromo ea kakaretso ea letoto le sa feleng: \[ S_{\infty} = \frac{a}{1 - r} = \frac{7}{1 - \frac{1}{3}} = \frac{7}{\frac{2}{3}} = 7 \cdot \frac{3}{2} = \frac{21}{2} = 10.5 \] Kahoo, kakaretso ea letoto le sa feleng ke 10.5. Mohlala Potso ea 4: Ho Fumana Melao le Litekanyetso tsa Potso ea Letoto: Kakaretso ea lipehelo tse 3 tsa pele tsa letoto la jeometri ke 21, 'me kakaretso ea lipehelo tsa 2 le 3 ke 18. Fumana hore na lentsoe la pele le karolelano ea lona ke lefe. Puisano: A re re nako ea pele ke \(a\) 'me karolelano ke \(r\). Ho tsoa tlhahisoleseding ea bothata, re ka ngola li-equation tse peli tse latelang: \[ a + ar + ar^2 = 21 \quad \text{(1)} \] \[ ar + ar^2 = 18 \quad \text{(2)} \] Ho tsoa ho equation (2), re ka hlalosa \(a\) ho latela \(r\): \[ a(r + r^2) = 18 \e bolela a = \frac{18}{r(1 + r)} \] E latelang, nkela \(a\) sebaka ka equation (1): \[ \frac{18(1)}{r(1 + r)} + \frac{18r}{r(1 + r)} + \frac{18r^2}{r(1 + r)} = 21 \] \[ \frac{18}{1 + r} + \frac{18r}{1 + r} + \frac{18r^2}{1 + r} = 21 \] \[ \frac{18 (1 + r + r^2)}{1 + r} = 21 \] \[ \frac{18 \cdot 3}{1 + r} = 21 \] \[ \frac{54}{1 + r} = 21 \] \[ 54 = 21(1 + r) \] \[ 54 = 21 + 21r \] \[ 33 = 21r \] \[ r = \frac{33}{21} = \frac{11}{7} \] Ha boleng ba \(r\) bo tsejwa, bo kgutlisetse boleng ba \(a\): \[ a = \frac{18}{r(1 + r)} = \frac{18}{\frac{11}{7} (1 + \frac{11}{7})} = \frac{18}{\frac{11}{7} \cdot \frac{18}{7}} = \frac{18 \cdot 7}{11 \cdot 18} = \frac{7}{11} \] Kahoo, lentsoe la pele \(a\) ke \(\frac{7}{11}\) mme karolelano ke \(\frac{11}{7}\). Qetello Letoto la jeometri ke mohopolo oa lipalo o sebelisoang haholo lits'ebetsong tse fapaneng. Ho utloisisa mekhoa ea motheo e kang lentsoe la nth, kakaretso ea mantsoe a pele a n, le kakaretso ea letoto la li-geometric tse sa feleng ke habohlokoa bakeng sa ho rarolla mathata a fapaneng a amanang le lipalo. Ka ho itloaetsa mathata a fapaneng a mehlala joalo ka ha ho tšohliloe sehloohong sena, re ka ntlafatsa bokhoni ba rona ba ho utloisisa le ho sebelisa letoto la libopeho tsa jeometri hamolemo.
