Ngā Tauira Pātai e Matapaki ana i ngā Raupapa Pāngarau
He ariā taketake ngā raupapatanga tau i roto i te pāngarau e puta pinepine ana i roto i ngā momo rapanga, i te kura tuarua me te mātauranga teitei. Kei roto i tēnei ariā he raupapatanga tau, ko ia tau he hua o te tāpiri, te tango rānei i tētahi tau pumau mai i te tau o mua. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira rapanga me ā rātou otinga hei mārama ake i te ariā o ngā raupapatanga tau.
Te Mārama ki ngā Raupapa Pāngarau
Ko te raupapa tātai he raupapa he rerekētanga pumau (rerekētanga) kei waenganui i ngā kupu e rua e whai ake nei. Hei tauira, mēnā ko te kupu tuatahi o te raupapa tātai he \(a\) me te rerekētanga \(d\), ka taea te tuhi i ngā kupu penei:
\[ a, a + d, a + 2d, a + 3d, \ldots \]
Ki te hiahia tātou ki te kimi i te tau tuawhā o tēnei raupapa, ko te tātai mō te tau tuawhā (\(U_n\)) koia tēnei:
\[ U_n = a + (n-1)d \]
I taua wā anō, ka taea te tatau i te tapeke o ngā kupu tuatahi n o tētahi raupapa tātai (\(S_n\)) mā te whakamahi i te tātai:
\[ S_n = \frac{n}{2} (2a + (n-1)d) \]
Ngā Pātai Tauira me te Kōrero
Tauira Pātai 1
Pātai: Hoatu he raupapa tātaitanga me te kupu tuatahi \(a = 5\) me te rerekētanga noa \(d = 3\). Kimihia te kupu 10 o te raupapa.
Kōrero:
Hei kimi i te wāhanga 10 (\(U_{10}\)), ka taea e tātou te whakamahi i te tātai wāhanga 9:
\[ U_{10} = a + (10-1)d \]
\[ U_{10} = 5 + (9 \cdot 3) \]
\[ U_{10} = 5 + 27 \]
\[ U_{10} = 32 \]
Nō reira, ko te 10 o ngā wāhanga o te raupapa ko te 32.
Tauira Pātai 2
Pātai: Kimihia te tapeke o ngā kupu tuatahi e 15 o te raupapa pāngarau, ko tōna kupu tuatahi ko \(a = 4\) ā, ko te rerekētanga noa ko \(d = 7\).
Kōrero:
Hei kimi i te tapeke o ngā kupu tuatahi e 15 (\(S_{15}\)), ka taea e tātou te whakamahi i te tātai mō te tapeke o ngā kupu tuatahi n:
\[ S_{15} = \frac{15}{2} (2a + (15-1)d) \]
\[ S_{15} = \frac{15}{2} (2 \cdot 4 + 14 \cdot 7) \]
\[ S_{15} = \frac{15}{2} (8 + 98) \]
\[ S_{15} = \frac{15}{2} \cdot 106 \]
\[ S_{15} = 15 \cdot 53 \]
\[ S_{15} = 795 \]
Nō reira, ko te tapeke o ngā kupu tuatahi e 15 o te raupapa ko te 795.
Tauira Pātai 3
Pātai: E mōhiotia ana ko te tau tuarima o tētahi raupapa tātai he 20, ā, ko te tau tuarima he 48. Kimihia te tau tuatahi (\(a\)) me te rerekētanga noa (\(d\)) o te raupapa.
Kōrero:
Mai i ngā tikanga kua hoatu:
\[ U_5 = a + 4d = 20 \]
\[ U_{12} = a + 11d = 48 \]
E rua ā tātou whārite rārangi me ngā taurangi e rua ka taea e tātou te whakaoti:
1. \( a + 4d = 20 \)
2. \( a + 11d = 48 \)
Mai i te whārite 1, ka taea e tātou te whakaatu i te \(a\) mā te whakamahi i te \(d\):
\[ a = 20 – 4d \]
Inaianei ka whakakapia te \(a\) ki te whārite 2:
\[ 20 – 4d + 11d = 48 \]
\[ 20 + 7d = 48 \]
\[ 7d = 28 \]
\[ d = 4 \]
Nā, ka whakakapia te uara o \(d\) ki te whārite \(a = 20 – 4d\):
\[ a = 20 – 4 \cdot 4 \]
\[ a = 20 – 16 \]
\[ a = 4 \]
Nō reira, ko te wāhanga tuatahi o te raupapa ko te 4, ā, ko te rerekētanga noa ko te 4.
Tauira Pātai 4
Pātai: E hia ngā kupu e hiahiatia ana mō tētahi raupapa pāngarau me te kupu tuatahi \(a = 2\) me te rerekētanga noa \(d = 5\) hei tapeke ki te 200?
Kōrero:
I tēnei wā, me kimi e tātou te tapeke o ngā kupu tuatahi n (\(S_n\)) e ōrite ana ki te 200. Whakamahia te tātai mō te tapeke o ngā kupu tuatahi n:
\[ S_n = \frac{n}{2} (2a + (n-1)d) = 200 \]
Whakakapia ngā uara o \(a\) me \(d\):
\[ \frac{n}{2} (2 \cdot 2 + (n-1) \cdot 5) = 200 \]
\[ \frac{n}{2} (4 + 5n – 5) = 200 \]
\[ \frac{n}{2} (5n – 1) = 200 \]
\[ n (5n – 1) = 400 \]
He whārite tapawhā tēnei. Hei whakaoti, ka whakarerekētia tōna āhua:
\[ 5n^2 – n – 400 = 0 \]
Whakamahia te tātai tapawhā \(ax^2 + bx + c = 0\):
\[ n = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
I tēnei wā \(a = 5\), \(b = -1\), me \(c = -400\):
\[ n = \frac{-(-1) \pm \sqrt{(-1)^2 – 4 \cdot 5 \cdot (-400)}}{2 \cdot 5} \]
\[ n = \frac{1 \pm \sqrt{1 + 8000}}{10} \]
\[ n = \frac{1 \pm \sqrt{8001}}{10} \]
E tata ana te uara o \(\sqrt{8001}\) ki te 89.42, kātahi:
\[ n = \frac{1 \pm 89.42}{10} \]
Ka tangohia e mātou ngā uara pai:
\[ n = \frac{1 + 89.42}{10} \]
\[ n \tata \frac{90.42}{10} \]
\[ n \tata ki te 9.042 \]
Nō reira, ko te maha o ngā kupu e hiahiatia ana ko te 9 ngā kupu (mēnā ka whakaawhiwhia).
Whakamutunga
He kaupapa nui ngā raupapatanga pāngarau i roto i te pāngarau. He mea tino nui te māramatanga hōhonu ki te kupu tuatahi, te rerekētanga noa, te kupu tuarima, me te tapeke o ngā kupu tuatahi n hei whakaoti rapanga maha. Mā te whakamahi i ngā tauira me ngā kōrero i runga ake nei, ko te tumanako ka pai ake te māramatanga o ngā kaipānui ki ngā ariā taketake o ngā raupapatanga pāngarau.
