Mehlala ea Lipotso Tse Buisanang ka Letoto la Lipalo
Tatellano ea lipalo ke khopolo ea motheo lipalong tse hlahang khafetsa mathateng a fapaneng, sekolong se phahameng le thutong e phahameng. Khopolo ena e kenyelletsa tatellano ea linomoro moo nako ka 'ngoe e leng sephetho sa ho eketsa kapa ho ntša palo e sa fetoheng ho tsoa kotareng e fetileng. Sehloohong sena, re tla buisana ka mathata a 'maloa a mehlala le litharollo tsa ona ho utloisisa haholoanyane khopolo ea tatellano ea lipalo.
Ho Utloisisa Letoto la Lipalo
Letoto la lipalo ke letoto le nang le phapang e sa fetoheng (phapang) pakeng tsa mareo a mabedi a latellanang. Mohlala, haeba letoto la lipalo le na le lentsoe la pele \(a\) le phapang \(d\), joale mareo a ka ngoloa ka tsela e latelang:
\[ a, a + d, a + 2d, a + 3d, \ldots \]
Haeba re batla ho fumana nako ya nth ya letoto lena, foromo ya nako ya nth (\(U_n\)) ke:
\[ U_n = a + (n-1)d \]
Ho sa le jwalo, kakaretso ya mantswe a pele a n a letoto la dipalo (\(S_n\)) e ka balwa ho sebediswa foromo:
\[ S_n = \frac{n}{2} (2a + (n-1)d) \]
Lipotso tsa Mehlala le Puisano
Mohlala oa Potso ea 1
Potso: Ho fanoe ka letoto la lipalo le nang le lentsoe la pele \(a = 5\) le phapang e tloaelehileng \(d = 3\). Fumana lentsoe la bo10 la letoto.
Puisano:
Ho fumana lentsoe la bo10 (\(U_{10}\)), re ka sebedisa foromo ya lentswe la bo10:
\[ U_{10} = a + (10-1)d \]
\[ U_{10} = 5 + (9 \cdot 3) \]
\[ U_{10} = 5 + 27 \]
\[ U_{10} = 32 \]
Kahoo, nako ea bo10 ea letoto lena ke 32.
Mohlala oa Potso ea 2
Potso: Fumana kakaretso ea mantsoe a 15 a pele a letoto la lipalo ao lentsoe la 'ona la pele e leng \(a = 4\) 'me phapang e tloaelehileng e le \(d = 7\).
Puisano:
Ho fumana kakaretso ea mantsoe a 15 a pele (\(S_{15}\)), re ka sebelisa foromo bakeng sa kakaretso ea mantsoe a pele a n:
\[ S_{15} = \frac{15}{2} (2a + (15-1)d) \]
\[ S_{15} = \frac{15}{2} (2 \cdot 4 + 14 \cdot 7) \]
\[ S_{15} = \frac{15}{2} (8 + 98) \]
\[ S_{15} = \frac{15}{2} \cdot 106 \]
\[ S_{15} = 15 \cdot 53 \]
\[ S_{15} = 795 \]
Kahoo, kakaretso ea lipehelo tse 15 tsa pele tsa letoto lena ke 795.
Mohlala oa Potso ea 3
Potso: Hoa tsebahala hore nako ea bo5 ea letoto la lipalo ke 20 'me nako ea bo12 ke 48. Fumana nako ea pele (\(a\)) le phapang e tloaelehileng (\(d\)) ea letoto.
Puisano:
Ho tsoa maemong a fanoeng:
\[ U_5 = a + 4d = 20 \]
\[ U_{12} = a + 11d = 48 \]
Re na le di-equation tse pedi tse otlolohileng tse nang le di-variable tse pedi tseo re ka di rarollang:
1. \( a + 4d = 20 \)
2. \( a + 11d = 48 \)
Ho tsoa ho equation ea 1, re ka hlalosa \(a\) ho latela \(d\):
\[a = 20 – 4d \]
Jwale re nkela \(a\) sebaka sa equation ya 2:
\[ 20 – 4d + 11d = 48 \]
\[ 20 + 7d = 48 \]
\[7d = 28 \]
\[d = 4 \]
Jwale re nkela boleng ba \(d\) sebaka ka equation \(a = 20 – 4d\):
\[a = 20 – 4 \cdot 4 \]
\[a = 20 – 16 \]
\[a = 4 \]
Kahoo, nako ea pele ea letoto ke 4 'me phapang e tloaelehileng ke 4.
Mohlala oa Potso ea 4
Potso: Ho hlokahala mantsoe a makae bakeng sa letoto la lipalo ka lentsoe la pele \(a = 2\) le phapang e tloaelehileng \(d = 5\) ho fihlela ho 200?
Puisano:
Tabeng ena, re hloka ho fumana kakaretso ea mantsoe a pele a n (\(S_n\)) a lekanang le 200. Sebelisa foromo bakeng sa kakaretso ea mantsoe a pele a n:
\[ S_n = \frac{n}{2} (2a + (n-1)d) = 200 \]
Kenya boleng ba \(a\) le \(d\):
\[ \frac{n}{2} (2 \cdot 2 + (n-1) \cdot 5) = 200 \]
\[ \frac{n}{2} (4 + 5n – 5) = 200 \]
\[ \frac{n}{2} (5n – 1) = 200 \]
\[ n (5n – 1) = 400 \]
Ena ke equation ea quadratic. Ho e rarolla, re fetola sebopeho sa eona:
\[ 5n^2 – n – 400 = 0 \]
Sebelisa foromo ea quadratic \(ax^2 + bx + c = 0\):
\[ n = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Tabeng ena \(a = 5\), \(b = -1\), le \(c = -400\):
\[ n = \frac{-(-1) \pm \sqrt{(-1)^2 – 4 \cdot 5 \cdot (-400)}}{2 \cdot 5} \]
\[ n = \frac{1 \pm \sqrt{1 + 8000}}{10} \]
\[ n = \frac{1 \pm \sqrt{8001}}{10} \]
Boleng ba \(\sqrt{8001}\) bo haufi le 89.42, ebe:
\[ n = \frac{1 \pm 89.42}{10} \]
Re nka litekanyetso tse ntle:
\[ n = \frac{1 + 89.42}{10} \]
\[ n \hoo e ka bang \frac{90.42}{10} \]
\[ n \hoo e ka bang 9.042 \]
Kahoo, palo ea mantsoe a hlokahalang ke mantsoe a 9 (haeba a phethotsoe).
Qetello
Tatellano ea lipalo ke sehlooho sa bohlokoa lipalo. Kutloisiso e batsi ea lentsoe la pele, phapang e tloaelehileng, lentsoe la nth, le kakaretso ea mantsoe a pele a n ke ea bohlokoa haholo bakeng sa ho rarolla mathata a mangata a fapaneng. Ka ho sebelisa mehlala le lipuisano tse kaholimo, ho tšeptjoa hore babali ba tla utloisisa hamolemo likhopolo tsa motheo tsa tatellano ea lipalo.
