Ngā Tauira Pātai e Matapaki ana i te Raupapa Āhuahanga Mutunga Kore
Ko te raupapa āhuahanga mutunga kore he raupapa e whakauru ana i te maha mutunga kore o ngā kupu i roto i te ahunga ā-āhuahanga. He whakaritenga motuhake kei roto i ēnei raupapa kia taea ai te tatau i tō rātou tapeke (te rohe rānei). I roto i tēnei tuhinga, ka matapakihia e mātou te ariā taketake o ngā raupapa āhuahanga mutunga kore, ngā whakaritenga mō te hanga i aua raupapa, me ētahi tauira rapanga me ō rātou otinga.
Ngā Ariā Taketake o te Raupapa Āhuahanga Mutunga Kore
Ko te tikanga, ko te raupapa ā-ira he raupapa tau e whiwhi ai i ia tau i muri i te tuatahi mā te whakarea i te tau o mua ki tētahi pūmau e kiia nei ko te ōwehenga noa (r). Me kī he raupapa ā-ira tā tātou:
\[ a, ar, ar^2, ar^3, ar^4, \ldots \]
Mō tētahi raupapa āhuahanga mutunga kore, ka whakaarohia e mātou te tapeke o ngā kupu katoa o te raupapa. Ko te tapeke o tēnei raupapa ka tautuhia penei:
\[ S = a + ar + ar^2 + ar^3 + ar^4 + \ldots \]
Ka tūtaki te tapeke o tētahi raupapa āhuahanga mutunga kore (he tapeke tūturu) mēnā ko te ōwehenga \( |r| < 1 \), ā, mēnā anake ko te ōwehenga \( |r| \geq 1 \). Mēnā \( |r| \geq 1 \), ka wehe te raupapa, ā, kāore he tapeke tūturu (ka tae ki te mutunga kore).
Mena ko \( |r| < 1 \), ka taea te tautuhi i te tapeke S o tētahi raupapa āhuahanga mutunga kore penei: \[ S = \frac{a}{1-r} \] ko: - \( S \) te tapeke o te raupapa, - \( a \) te kupu tuatahi, - \( r \) te ōwehenga. Ngā Tauira Pātai me te Kōrero Tauira Pātai 1 Pātai: Kimihia te tapeke o te raupapa āhuahanga mutunga kore mō te raupapa e whai ake nei: \[ 3 + 1.5 + 0.75 + 0.375 + \ldots \] Kōrero: Me tautuhi tātou i ngā huānga nui o te raupapa: Ko te kupu tuatahi \( a = 3 \) Ka taea te kimi i te ōwehenga \( r \) mā te wehewehe i te kupu tuarua ki te kupu tuatahi, arā: \[ r = \frac{1.5}{3} = 0.5 \] Nā te mea ko \( |r| = 0.5 < 1 \), ka tūhono tēnei raupapa, ā, ka taea e tātou te tatau i te tapeke o te raupapa mutunga kore. Whakamahia te tātai mō te tapeke o tētahi raupapa āhuahanga mutunga kore: \[ S = \frac{a}{1-r} \] \[ S = \frac{3}{1-0.5} \] \[ S = \frac{3}{0.5} \] \[ S = 6 \] Nō reira, ko te tapeke o te raupapa āhuahanga mutunga kore ko te 6. Tauira Pātai 2 Pātai: Kimihia te tapeke o tētahi raupapa āhuahanga mutunga kore me te tau tuatahi ko te 8 me te ōwehenga \( r = -\frac{1}{3} \). Kōrero: Ko te kupu tuatahi \( a = 8 \) Ōwehenga \( r = -\frac{1}{3} \) Nā te mea \( |r| = \frac{1}{3} < 1 \), ka tūhono tēnei raupapa, ā, ka taea e tātou te tatau i te tapeke o te raupapa mutunga kore. Whakamahia te tātai mō te tapeke o tētahi raupapa āhuahanga mutunga kore: \[ S = \frac{a}{1-r} \] \[ S = \frac{8}{1 - \left(-\frac{1}{3}\right)} \] \[ S = \frac{8}{1 + \frac{1}{3}} \] \[ S = \frac{8}{\frac{4}{3}} \] \[ S = 8 \times \frac{3}{4} \] \[ S = 6 \] Nō reira, ko te tapeke o te raupapa āhuahanga mutunga kore ko te 6. Tauira Pātai 3 Pātai: He tapeke mutunga kore tō te raupapa e whai ake nei? Mēnā koinā, kimihia te tapeke. \[ 5 + 2.5 + 1.25 + 0.625 + \ldots \] Kōrero: Ko te kupu tuatahi \( a = 5 \) Ka taea te kimi i te ōwehenga \( r \) mā te wehewehe i te kupu tuarua ki te kupu tuatahi, arā: \[ r = \frac{2.5}{5} = 0.5 \] Nā te mea \( |r| = 0.5 < 1 \), ka tūhono tēnei raupapa, ā, ka taea e tātou te tatau i te tapeke o te raupapa mutunga kore. Whakamahia te tātai mō te tapeke o tētahi raupapa āhuahanga mutunga kore: \[ S = \frac{a}{1-r} \] \[ S = \frac{5}{1-0.5} \] \[ S = \frac{5}{0.5} \] \[ S = 10 \] Nō reira, ko te tapeke o te raupapa āhuahanga mutunga kore ko te 10. Tauira Pātai 4 Pātai: Whakatauhia mēnā he tūhono, he rerekē rānei te raupapa e whai ake nei: \[ 4 - 6 + 9 - 13.5 + \ldots \] Kōrero: Ko te kupu tuatahi \( a = 4 \) Ka kitea te ōwehenga \( r \) mā te wehewehe i te kupu tuarua ki te kupu tuatahi, arā: \[ r = \frac{-6}{4} = -1.5 \] Nā te mea \( |r| = 1.5 > 1 \), he rerekē tēnei raupapa, ā, kāore he tapeke tuturu.Nō reira, he rerekē te raupapa.
Tauira Pātai 5
Pātai: Me kī kei a koe te raupapa mutunga kore e whai ake nei:
\[ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots \]
Whakatauhia te tapeke o te raupapa.
Kōrero:
Te wāhanga tuatahi \( a = \frac{1}{2} \)
Ka kitea te ōwehenga \( r \) mā te wehewehe i te wāhanga tuarua ki te wāhanga tuatahi, arā:
\[ r = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2} \]
Mai i te mea \( |r| = \frac{1}{2} < 1 \), ka tūhono tēnei raupapa, ā, ka taea e tātou te tatau i te tapeke o te raupapa mutunga kore. Whakamahia te tātai mō te tapeke o tētahi raupapa āhuahanga mutunga kore: \[ S = \frac{a}{1-r} \] \[ S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} \] \[ S = \frac{\frac{1}{2}}{\frac{1}{2}} \] \[ S = 1 \] Nō reira, ko te tapeke o te raupapa āhuahanga mutunga kore ko te 1. Whakamutunga He ariā pāngarau nui te raupapa āhuahanga mutunga kore e whakamahia whānuitia ana i roto i ngā mara maha. Hei whakatau i te tapeke o tētahi raupapa āhuahanga mutunga kore, me whakarite kia rite te ōwehenga \( |r| < 1 \). Nō reira, ka taea te tatau i te tapeke o te raupapa mā te whakamahi i tētahi tātai māmā, tika hoki. Mai i ngā tauira raruraru i runga ake nei, ka kite tātou he tino māmā te whakaoti rapanga e pā ana ki ngā raupapa āhuahanga mutunga kore nā tēnei tikanga.
