Ngā Tauira Pātai e Matapaki ana i ngā Raupapa Pāngarau
Ko ngā raupapatanga tātai he ariā taketake i roto i te pāngarau e puta pinepine ana i roto i ngā whakamātautau me ngā tono o te ao tūturu. Ko te raupapatanga tātai he raupapatanga tau me te rerekētanga pumau i waenga i ia rua ngā kupu karapīpiti. I roto i tēnei tuhinga, ka tūhuratia e mātou te ariā o ngā raupapatanga tātai mā roto i ētahi tauira rapanga me ngā whakamārama whānui.
Ngā Whakamāramatanga me ngā Tuhituhinga
I mua i te urunga atu ki ngā tauira rapanga, he mea nui kia mārama ki te tuhi e whakamahia ana i roto i ngā raupapatanga tātai. Mena ko \(a\) te kupu tuatahi, ā, ko \(d\) te rerekētanga noa (rerekētanga pumau), ka taea te tuhi i te raupapatanga tātai penei:
\[ a, a+d, a+2d, a+3d, \ldots, a+(n-1)d \]
Ka taea te whakatakoto i te kupu 9 (Un) o tēnei raupapatanga penei:
\[ U_n = a + (n-1)d \]
Anei ētahi tauira pātai me ā rātou matapakinga hei whakamāmā ake i te mārama ki ngā raupapatanga pāngarau.
Tauira Pātai 1
Pātai:
Hoatu he raupapatanga tātai me te kupu tuatahi \(a = 5\) me te rerekētanga noa \(d = 3\). Whakatauhia te kupu 10 o te raupapatanga.
Kōrero:
Mā te whakamahi i te tātai whānui mō te kupu tuawhā, arā, \( U_n = a + (n-1)d \):
\[ U_{10} = 5 + (10-1) \cdot 3 \]
\[ U_{10} = 5 + 9 \cdot 3 \]
\[ U_{10} = 5 + 27 \]
\[ U_{10} = 32 \]
Nō reira, ko te 10 o ngā wāhanga o te raupapatanga ko te 32.
Tauira Pātai 2
Pātai:
Hoatu he raupapatanga pāngarau me te tau tuarima he 20, me te tau tuawaru he 35. Whakatauhia te tau tuatahi \(a\) me te rerekētanga noa \(d\) o te raupapatanga.
Kōrero:
Mai i te pātai, e mōhio ana tātou:
\[ U_5 = a + 4d = 20 \]
\[ U_8 = a + 7d = 35 \]
Tangohia ngā whārite e rua hei whakakore i te \(a\):
\[ (a + 7d) – (a + 4d) = 35 – 20 \]
\[ 3d = 15 \]
\[ d = 5 \]
Kātahi ka whakakapia te \(d = 5\) hei kimi i te \(a\):
\[ a + 4 \cdot 5 = 20 \]
\[ a + 20 = 20 \]
\[ a = 0 \]
Nō reira, ko te tau tuatahi o te raupapatanga ko te 0, ā, ko te rerekētanga ko te 5.
Tauira Pātai 3
Pātai:
He aha te tapeke o ngā kupu tuatahi e 20 o tētahi raupapatanga pāngarau ko tōna kupu tuatahi ko \(a = 2\) me te rerekētanga noa \(d = 4\)?
Kōrero:
Ka taea te tatau i te tapeke o ngā kupu tuatahi n o tētahi raupapatanga taupū mā te whakamahi i te tātai:
\[ S_n = \frac{n}{2} \left( 2a + (n-1)d \right) \]
Mō tēnei rārangi:
\[ S_{20} = \frac{20}{2} \left( 2 \cdot 2 + (20-1) \cdot 4 \right) \]
\[ S_{20} = 10 \left( 4 + 76 \right) \]
\[ S_{20} = 10 \cdot 80 \]
\[ S_{20} = 800 \]
Nō reira, ko te tapeke o ngā kupu tuatahi e 20 o te raupapatanga ko 800.
Tauira Pātai 4
Pātai:
Ko te tuatoru o ngā wāhanga o tētahi raupapatanga pāngarau ko te 15, ā, ko te tuawhitu o ngā wāhanga ko te 27. Kimihia te tekau mā rua o ngā wāhanga o te raupapatanga.
Kōrero:
Tuatahi, me kimi e tātou ngā uara \(a\) me \(d\). Mai i te pātai, e mōhio ana tātou:
\[ U_3 = a + 2d = 15 \]
\[ U_7 = a + 6d = 27 \]
Tangohia ngā whārite e rua hei whakakore i te \(a\):
\[ (a + 6d) – (a + 2d) = 27 – 15 \]
\[ 4d = 12 \]
\[ d = 3 \]
Kātahi ka whakakapia te \(d = 3\) hei kimi i te \(a\):
\[ a + 2 \cdot 3 = 15 \]
\[ a + 6 = 15 \]
\[ a = 9 \]
Mā te whakamahi i te tātai o te wāhanga 9 hei kimi i te wāhanga 12:
\[ U_{12} = a + 11d \]
\[ U_{12} = 9 + 11 \cdot 3 \]
\[ U_{12} = 9 + 33 \]
\[ U_{12} = 42 \]
Nō reira, ko te 12 o ngā wāhanga o te raupapatanga ko te 42.
Tauira Pātai 5
Pātai:
Ko te raupapatanga pāngarau me te kupu tuatahi \(a\) me te rerekētanga noa \(d\) he tapeke o ngā kupu tuatahi e 10, arā, 55. Mena \(d = 1\), whakatauhia te kupu tuatahi \(a\).
Kōrero:
I hoatu ko \(d = 1\) me \(S_{10} = 55\). Whakamahia te tātai mō te tapeke o ngā kupu tuatahi:
\[ S_n = \frac{n}{2} (2a + (n-1)d) \]
Mō n = 10:
\[ S_{10} = \frac{10}{2} (2a + 9 \cdot 1) = 55 \]
\[ 5 (2a + 9) = 55 \]
\[ 2a + 9 = 11 \]
\[ 2a = 2 \]
\[ a = 1 \]
Nō reira, ko te tau tuatahi o te raupapatanga ko 1.
Whakamutunga
He ariā taketake ngā raupapatanga taunga i roto i te pāngarau, ā, he tino whai hua i roto i ngā momo mara. I roto i tēnei tuhinga, kua matapakihia e mātou ētahi tauira o ngā raupapatanga taunga me ō rātou otinga. Mā te mārama pai ki ngā tātai me ngā āhuatanga taketake o ngā raupapatanga taunga ka tino āwhina i te whakaoti rapanga rerekē e pā ana ki tēnei kaupapa.
Mā te whakaharatau i ngā tauira pēnei i ngā mea o runga ake nei, ko te tūmanako ka nui ake tō pūkenga me tō tere ake ki te whakaoti rapanga e pā ana ki ngā raupapatanga pāngarau.
