Ngā Tauira Pātai mō te Matapaki i ngā Raupapa Āhuahanga
He ariā nui ngā raupapa āhuahanga i roto i te pāngarau, e puta pinepine ana i roto i ngā momo raruraru, tae atu ki ngā whakamātautau kura, ngā whakamātautau whakauru ki te whare wānanga, tae atu ki ngā whakamātautau paerewa pērā i te SAT, te GRE rānei. Mā te mārama hohonu ki ngā raupapa āhuahanga ka āwhina i a tātou ki te whakaoti rapanga me te whai hua. Ka kapi tēnei tuhinga i ētahi tauira raruraru, ā, ka matapakihia hoki ngā raupapa āhuahanga.
Te Mārama ki ngā Raupapa Āhuahanga
Ko te raupapa ā-ira he raupapa e whiwhihia ai ia wāhanga mā te whakarea i te wāhanga o mua ki tētahi tau pumau e kiia nei ko te ōwehenga (ōwehenga noa, e tohuhia ana e te reta \(r\)). I te nuinga o te wā, ka taea te tuhi i tētahi raupapa ā-ira penei:
\[
a, ar, ar^2, ar^3, \ldots
\]
Kei hea:
– Ko te kupu tuatahi ko \(a\)
– Ko te ōwehenga o te raupapa ko te \(r\)
Mena \( |r| < 1 \), he āhuatanga whakamere o ngā raupapa āhuahanga mutunga kore. He maha ngā whakamahinga mahi o ngā raupapa ā-ira i roto i ngā momo mara pērā i te ahupūngao, te ōhanga, me te koiora. Tātai Raupapa Āhuahanga Te wāhanga 9 o tētahi raupapa āhuahanga Ka taea te tatau i te wāhanga 9 o tētahi raupapa āhuahanga mā te whakamahi i te tātai: \[ U_n = a \cdot r^{n-1} \] Tapeke o ngā Wāhanga n Tuatahi o tētahi Raupapa Āhuahanga Ka taea te tatau i te tapeke o ngā \(n\) ngā wāhanga tuatahi o tētahi raupapa āhuahanga (Sn) mā te whakamahi i te tātai: \[ S_n = a \frac{1 - r^n}{1 - r}, \quad \text{for } r \neq 1 \] \[ S_n = na, \quad \text{for } r = 1 \] Tapeke Mutunga Kore o tētahi Raupapa Āhuahanga Mena \(|r| < 1\), ko te tapeke o tētahi raupapa āhuahanga mutunga kore: \[ S_{\infty} = \frac{a}{1 - r} \] Tauira Pātai me ngā Kōrero Ko ētahi tauira o ngā pātai raupapa āhuahanga me ā rātou kōrero: Tauira Pātai 1: Te tatau i te Pātai Wāhanga 9: Hoatu he raupapa āhuahanga me te wāhanga tuatahi \(a = 5\) me te ōwehenga \(r = 3\). Tātaihia te wāhanga tuaono o te raupapa. Kōrero: Mā te whakamahi i te tātai o te kupu 9: \[ U_6 = a \cdot r^{(6-1)} = 5 \cdot 3^5 = 5 \cdot 243 = 1215 \] Nō reira, ko te kupu 6 o te raupapa ko 1215. Tauira Pātai 2: Te Tatau i te Tapeke o ngā Kupu-n Tuatahi Pātai: Tātaihia te tapeke o ngā kupu-4 tuatahi o tētahi raupapa āhuahanga me te kupu-tuatahi \(a = 2\) me te ōwehenga \(r = \frac{1}{2}\). Kōrero: Mā te whakamahi i te tātai mō te tapeke o ngā kupu tuatahi \(n\): \[ S_4 = a \frac{1 - r^4}{1 - r} = 2 \frac{1 - (\frac{1}{2})^4}{1 - \frac{1}{2}} = 2 \frac{1 - \frac{1}{16}}{\frac{1}{2}} = 2 \frac{\frac{15}{16}}{\frac{1}{2}} = 2 \cdot \frac{15}{8} = 2 \cdot \frac{15}{8} = \frac{30}{8} = 3.75 \] Nō reira, ko te tapeke o ngā kupu tuatahi e 4 o te raupapa he 3.75. Tauira Pātai 3: Te Tapeke o tētahi Raupapa Āhuahanga Mutunga Kore Pātai: Tātaihia te tapeke o tētahi raupapa mutunga kore ina ko \(a = 7\) me \(r = \frac{1}{3}\). Kōrero: Mā te whakamahi i te tātai mō te tapeke o tētahi raupapa mutunga kore: \[ S_{\infty} = \frac{a}{1 - r} = \frac{7}{1 - \frac{1}{3}} = \frac{7}{\frac{2}{3}} = 7 \cdot \frac{3}{2} = \frac{21}{2} = 10.5 \] Nō reira, ko te tapeke o te raupapa mutunga kore ko te 10.5. Tauira Pātai 4: Te Whakatau i ngā Kupu me ngā Tauwehenga o tētahi Raupapa Pātai: Ko te tapeke o ngā kupu tuatahi e 3 o tētahi raupapa āhuahanga ko te 21, ā, ko te tapeke o ngā kupu tuarua me te tuatoru ko te 18. Whakatauhia te wāhanga tuatahi me tōna ōwehenga. Kōrero: Me kī ko te kupu tuatahi ko \(a\) ā, ko te ōwehenga ko \(r\). Mai i ngā mōhiohio raruraru, ka taea e tātou te tuhi i ngā whārite e rua e whai ake nei: \[ a + ar + ar^2 = 21 \quad \text{(1)} \] \[ ar + ar^2 = 18 \quad \text{(2)} \] Mai i te whārite (2), ka taea e tātou te whakaatu i te \(a\) i roto i ngā kupu o \(r\): \[ a(r + r^2) = 18 \implies a = \frac{18}{r(1 + r)} \] Muri iho, whakakapia te \(a\) ki te whārite (1): \[ \frac{18(1)}{r(1 + r)} + \frac{18r}{r(1 + r)} + \frac{18r^2}{r(1 + r)} = 21 \] \[ \frac{18}{1 + r} + \frac{18r}{1 + r} + \frac{18r^2}{1 + r} = 21 \] \[ \frac{18 (1 + r + r^2)}{1 + r} = 21 \] \[ \frac{18 \cdot 3}{1 + r} = 21 \] \[ \frac{54}{1 + r} = 21 \] \[ 54 = 21(1 + r) \] \[ 54 = 21 + 21r \] \[ 33 = 21r \] \[ r = \frac{33}{21} = \frac{11}{7} \] Kia mōhiotia te uara o \(r\), whakakapia ki te uara o \(a\): \[ a = \frac{18}{r(1 + r)} = \frac{18}{\frac{11}{7} (1 + \frac{11}{7})} = \frac{18}{\frac{11}{7} \cdot \frac{18}{7}} = \frac{18 \cdot 7}{11 \cdot 18} = \frac{7}{11} \] Nō reira, ko te kupu tuatahi \(a\) ko \(\frac{7}{11}\) ā, ko te ōwehenga ko \(\frac{11}{7}\). Whakatau He ariā pāngarau te raupapa ā-ira e whakamahia whānuitia ana i roto i ngā momo tono. He mea nui te mārama ki ngā tātai taketake pēnei i te kupu tuawhā, te tapeke o ngā kupu tuatahi n, me te tapeke o tētahi raupapa āhuahanga mutunga kore hei whakaoti rapanga pāngarau e pā ana. Mā te whakaharatau i ngā tauira rapanga kua kōrerotia i roto i tēnei tuhinga, ka taea e tātou te whakapakari ake i tō tātou mārama me te whakamahi i ngā raupapa āhuahanga.
