Raupapa Āhuahanga Mutunga Kore: He Tūhuratanga Pāngarau
Pendahuluan
I roto i te pāngarau, he mea nui te ariā o te raupapa, i roto i ngā tono mahi me te māramatanga ariā. Ko tētahi momo raupapa whakamere hei ako ko ngā raupapa āhuahanga, inā koa, ko ngā raupapa āhuahanga mutunga kore, he āhuatanga ahurei, he whakamīharo hoki ō rātou. Ka tūhuratia e tēnei tuhinga ngā ariā, ngā āhuatanga, me ngā tono taketake o ngā raupapa āhuahanga mutunga kore, me te whakarato hoki i te māramatanga ki te āhua o te puta o ēnei raupapa i roto i ngā momo mara pūtaiao.
Te Whakamāramatanga o te Raupapa Āhuahanga
I te nuinga o te wā, ko te raupapa āhuahanga he raupapa e whiwhi ai i ia wāhanga i muri i te tuatahi mā te whakarea i te wāhanga o mua ki tētahi tau pumau e kiia nei ko te ōwehenga (r). Mena ko \( a \) te wāhanga tuatahi, ā, ko \( r \) te ōwehenga, ko te āhua whānui o te raupapa āhuahanga ko:
\[ a, ar, ar^2, ar^3, ar^4, \ldots \]
Ina whakaarohia he raupapa ā-ira mutunga kore, e kōrero ana tātou mō te tapeke o ngā kupu e haere tonu ana mō ake tonu atu.
Te Huinga o ngā Raupapa Āhuahanga Mutunga Kore
Ko tētahi āhuatanga whakamere o tētahi raupapa āhuahanga mutunga kore ko te mea ka whai tapeke mutunga kore (ka hono rānei) mēnā kei waenganui i te -1 me te 1 te ōwehenga \( r \) (arā, \(|r| < 1\)). Hei mārama ki te take o tēnei, ka taea e tātou te titiro ki ngā āhuatanga taketake o ngā tapeke raupapa mutunga kore. Hei tauira, me whakaaro tātou ki tētahi raupapa ā-ira mutunga kore me te kupu tuatahi \( a \) me te ōwehenga noa \( r \): \[ S = a + ar + ar^2 + ar^3 + \ldots \] Mēnā ka whakareatia ia kupu ki te ōwehenga noa \( r \), ka taea e tātou te tuhi: \[ rS = ar + ar^2 + ar^3 + ar^4 + \ldots \] Hei kimi i te tapeke o tēnei raupapa, tangohia te whārite tuarua mai i te tuatahi: \[ S - rS = a \] Kātahi ka taea e tātou te tauwehe \( S \) mai i te whārite: \[ S(1 - r) = a \] Pēnei: \[ S = \frac{a}{1 - r} \] He tika tēnei tātai mēnā ko \(|r| < 1\). Mena ko \(|r| \geq 1\), e kore te raupapa e tūtaki nā te mea ka nui haere ngā kupu, ka wiriwiri rānei me te kore here. Ngā Tauira o te Raupapa Āhuahanga Mutunga Kore Me arotake ētahi tauira hei whakarato i tētahi horopaki mārama ake mō tēnei ariā. 1. Raupapa Āhuahanga Māmā Whakaarohia he raupapa ā-irahanga me \( a = 1 \) me \( r = \frac{1}{2} \): \[ 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \] Mā te whakamahi i te tātai mō te tapeke o tētahi raupapa ā-irahanga mutunga kore, ka kitea e tātou te tapeke: \[ S = \frac{1}{1 - \frac{1}{2}} = 2 \] Nō reira, ko te tapeke o tēnei raupapa ko te 2. 2. Raupapa Āhuahanga me te Ōwehenga Kino Whakaarohia te raupapa me \( a = 3 \) me \( r = -\frac{1}{3} \): \[ 3 - 1 + \frac{1}{3} - \frac{1}{9} + \ldots \] Ka taea te tatau i te tapeke o tēnei raupapa mā te whakamahi i te tātai kotahi: \[ S = \frac{3}{1 - (-\frac{1}{3})} = \frac{3}{1 + \frac{1}{3}} = \frac{3}{\frac{4}{3}} = \frac{3 \times 3}{4} = \frac{9}{4} = 2.25 \] Ngā Whakamahinga o te Raupapa Āhuahanga Mutunga Kore He whānuitia ngā whakamahinga o ngā raupapa āhuahanga mutunga kore i roto i te pūtaiao me te hangarau. Ko ētahi tauira: 1. Te Pūtea me te Ōhanga I roto i te pūtea, he maha ngā wā ka whakamahia ngā raupapa ā-ira mutunga kore i roto i ngā ariā o te uara o nāianei me te uara ā-muri o tētahi utu ā-tau. Mena ka whiwhi te tangata i tētahi utu pumau e mau tonu ana mō ake tonu atu, ko te uara o nāianei o taua utu ko te tapeke o tētahi raupapa āhuahanga mutunga kore. 2. Ahupūngao I roto i te ahupūngao, ka puta ake ngā raupapa ā-ira i roto i te tatau i ngā āhuatanga orooro me te whakaohooho i roto i ngā pūnaha ā-tinana. Ko tētahi tauira matarohia ko te ine i te roa o te mata o tētahi waea e tapahia ana i tētahi ōwehenga. 3. Pūtaiao Rorohiko I roto i te pūtaiao rorohiko, ko ētahi rauropi e whakamahi ana i ngā huarahi auau, auau rānei, ka whakamahi i ngā mātāpono mai i ngā raupapa ā-ira hei tātaritanga uauatanga wā. 4. Ngā Kōwhiringa Pūtea Ka whakamahia hoki ngā raupapa āhuahanga i roto i te whakatauira utu kōwhiringa, inā koa i roto i ngā tikanga pēnei i te tauira rua-ira i roto i te utu kōwhiringa, he taputapu nui tēnei i roto i te pūtea pāngarau. Ngā Āhuatanga Kē Atu o te Raupapa Āhuahanga Haunga te pūtahitanga, he maha atu anō ngā āhuatanga whakamere o te raupapa āhuahanga. Ko tētahi o ēnei ko ngā tauira o te wehewehe me te tāruarua, ā, e whai tikanga ai tēnei i roto i te toi, te hoahoanga, tae noa ki te puoro. E whakamahia ana hoki ngā raupapa ā-ira i roto i te tātari raupapa wā me te whakatauira tūponotanga i roto i ngā tatauranga. Whakamutunga Ko te raupapa ā-ira mutunga kore tētahi o ngā ariā nui o te pāngarau, ā, he maha ngā tono e horapa ana ki ngā momo mara pūtaiao. Ko te mārama ki ngā āhuatanga o te huihuinga me te āheinga ki te tatau i te tapeke o ēnei raupapa he taputapu tino kaha mō ngā kaipūtaiao, ngā miihini, ngā tohunga ōhanga, me ngā kaimahi i roto i ngā momo marautanga. Mā roto i tēnei ariā, ka kitea te ātaahua me te whakamere o te pāngarau i roto i te whakamārama i ngā āhuatanga o te ao tūturu i roto i tētahi huarahi tino pūnaha, tino arorau hoki. Mai i te ariā ki te mahi, ko te raupapa ā-ira mutunga kore tētahi o ngā pou matua o te ako i te pāngarau me ōna tono i te oranga o ia rā.
