Mehlala ea Lipotso Tse Buisanang ka Letoto la Lijiometri
Tatellano ea jeometri ke mohopolo oa motheo lipalong tse atisang ho rutoa sekolong se phahameng. Tatellano ena e na le sete ea linomoro, tseo e 'ngoe le e 'ngoe ea tsona e leng sehlahisoa sa nomoro e fetileng ka ntho e sa fetoheng e tsejoang e le "karolelano." Sehlooho sena se tla akaretsa mathata a 'maloa a mehlala le lipuisano tsa tatellano ea jeometri, ka tšepo ea ho thusa babali ho utloisisa mohopolo ona hamolemo.
Ho Utloisisa Letoto la Lijiometri
Tatellano ea jeometri ke tatelano ea linomoro tse entsoeng ka ho atisa nomoro ea pele (a) ka karolelano e tsitsitseng (r). Ka kakaretso, sebopeho se akaretsang sa tatelano ea jeometri ke:
\[ a, ar, ar^2, ar^3, \ldots, ar^{n-1} \]
Mona:
– “a” ke lentsoe la pele la tatellano.
– “r” ke karolelano ya lentswe le le leng ho lentswe le fetileng.
– “n” ke lentsoe la bohlano tatellanong.
Lipotso tsa Mehlala le Puisano
A re buisaneng ka mehlala e meng ea mathata ho utloisisa haholoanyane ka tatellano ea jeometri.
Mohlala oa Potso ea 1
Potso:
Ha ho fanoe ka tatellano ea jeometri, lentsoe la pele (a) ke 3 'me karolelano (r) ke 2. Fumana:
1. Temana ea bo5 ea tatellano.
2. Kakaretso ea mantsoe a 6 a pele a tatellano.
Puisano:
1. Teremo ea bo5 (U5) e ka baloa ho sebelisoa foromo ea teremo ea bo-nth ea tatellano ea jeometri, e leng:
\[ U_n = a \cdot r^{n-1} \]
Ho kenya a = 3, r = 2, le n = 5 foromong ena:
\[ U_5 = 3 \cdot 2^{5-1} \]
\[ U_5 = 3 \cdot 2^4 \]
\[ U_5 = 3 \cdot 16 \]
\[ U_5 = 48 \]
Kahoo, nako ea bo5 ea tatellano ke 48.
2. Kakaretso ea mantsoe a 6 a pele (S6) a tatellano ea jeometri e ka baloa ho sebelisoa foromo ea kakaretso ea mantsoe a pele a n, e leng:
\[ S_n = a \left( \frac{r^n – 1}{r – 1} \right) \]
Ho kenya a = 3, r = 2, le n = 6 foromong ena:
\[ S_6 = 3 \left( \frac{2^6 – 1}{2 – 1} \right) \]
\[ S_6 = 3 \left( \frac{64 – 1}{1} \right) \]
\[ S_6 = 3 \ka letsohong le letšehali( 63 \ka letsohong le letona) \]
\[ S_6 = 189 \]
Kahoo, kakaretso ea mantsoe a 6 a pele a tatellano ke 189.
Mohlala oa Potso ea 2
Potso:
Tatellano ea jeometri e na le nako ea boraro ea 27 le nako ea bohlano ea 243. Fumana boleng ba nako ea pele (a) le karolelano (r).
Puisano:
Ho fanoe ka U3 = 27 le U5 = 243. Ho sebelisoa foromo bakeng sa nako ea bo-nth ea tatellano ea jeometri:
\[ U_n = a \cdot r^{n-1} \]
Bakeng sa U3:
\[ U_3 = a \cdot r^2 \]
\[ 27 = a \cdot r^2 \] \[ (1) \]
Bakeng sa U5:
\[ U_5 = a \cdot r^4 \]
\[ 243 = a \cdot r^4 \] \[ (2) \]
Ho bapisa diekate (1) le (2) ho tlosa a:
\[ \frac{U_5}{U_3} = \frac{a \cdot r^4}{a \cdot r^2} \]
\[ \frac{243}{27} = r^2 \]
\[9 = r^2 \]
\[r = 3 \mongolo{ kapa } r = -3 \]
Kenya boleng ba r ho equation (1):
Haeba \(r = 3 \):
\[ 27 = a \cdot 3^2 \]
\[ 27 = a \cdot 9 \]
\[a = 3 \]
Haeba \(r = -3 \):
\[ 27 = a \cdot (-3)^2 \]
\[ 27 = a \cdot 9 \]
\[a = 3 \]
Kahoo, lentsoe la pele (a) ke 3, 'me karolelano (r) e ka ba 3 kapa -3.
Mohlala oa Potso ea 3
Potso:
Fumana kakaretso e sa feleng ea letoto le latelang la jeometri haeba lentsoe la pele (a) e le 8 'me karolelano (r) e le 1/2.
Puisano:
Kakaretso e sa feleng ea letoto la jiometri e ka baloa ho sebelisoa foromo:
\[ S_{\infty} = \frac{a}{1 – r} \]
Ho kenya a = 8 le r = 1/2 foromong:
\[ S_{\infty} = \frac{8}{1 – \frac{1}{2}} \]
\[ S_{\infty} = \frac{8}{\frac{1}{2}} \]
\[ S_{\infty} = 8 \makgetlo a 2 \]
\[ S_{\infty} = 16 \]
Kahoo, kakaretso e sa feleng ea letoto la li-geometric ke 16.
Mohlala oa Potso ea 4
Potso:
Tatellano ea jeometri e na le nako ea bobeli ea 12 le nako ea bone ea 108. Fumana karolelano le nako ea pele ea tatellano.
Puisano:
Ho fanoe \( U_2 = 12 \) le \( U_4 = 108 \). Ho sebelisoa foromo bakeng sa nako ea bo-nth ea tatellano ea jeometri:
Bakeng sa \( U_2 \):
\[ U_2 = a \cdot r \]
\[ 12 = a \cdot r \] \[ (1) \]
Bakeng sa \( U_4 \):
\[ U_4 = a \cdot r^3 \]
\[ 108 = a \cdot r^3 \] \[ (2) \]
Ho bapisa diekate (1) le (2) ho tlosa a:
\[ \frac{U_4}{U_2} = \frac{a \cdot r^3}{a \cdot r} \]
\[ \frac{108}{12} = r^2 \]
\[9 = r^2 \]
\[r = 3 \mongolo{ kapa } r = -3 \]
Kenya boleng ba r ho equation (1):
Haeba \(r = 3 \):
\[ 12 = a \cdot 3 \]
\[a = 4 \]
Haeba \(r = -3 \):
\[ 12 = a \cdot (-3) \]
\[ a = -4 \]
Kahoo, lentsoe la pele (a) e ka ba 4 kapa -4, 'me karolelano (r) e ka ba 3 kapa -3.
Qetello
Tatellano ea jeometri ke mohopolo oa bohlokoa oa lipalo o sebelisoang khafetsa masimong a fapaneng. Ka ho utloisisa metheo le ho itloaetsa bokhoni ba ho rarolla mathata, re tšepa hore re ka ba le boiphihlelo bo eketsehileng ba ho utloisisa le ho sebelisa mohopolo. Sengoloa sena se kenyelletsa mehlala e 'maloa ea mathata le lipuisano ho thusa babali ho ithuta le ho utloisisa tatellano ea jeometri ka botebo. Re tšepa hore sena se tla thusa!
