Raupapa Āhuahanga: Ariā, Ngā Whakamahinga, me Ngā Tauira
Pendahuluan
He ariā taketake ngā raupapatanga ā-ira i roto i te pāngarau, ā, he whānuitia ngā tono i roto i ngā momo mara, tae atu ki te ōhanga, te ahupūngao, te koiora, me te hangarau. I roto i tēnei tuhinga, ka matapakihia e mātou te whakamāramatanga, ngā āhuatanga, me ngā tono o ngā raupapatanga ā-ira, me ētahi tauira hei whakamārama i tō mātou māramatanga.
Te Whakamāramatanga o te Raupapa Āhuahanga
Ko te raupapatanga ā-ira he raupapatanga e whiwhihia ai ia kupu i muri i te tuatahi mā te whakarea i te kupu o mua ki tētahi pūmau e kiia nei ko te ōwehenga noa (e tohuhia ana e r). I te nuinga o te wā, mēnā ko \(a_1\) te kupu tuatahi o te raupapatanga, ka taea te whakaatu i ngā kupu e whai ake nei hei \(a_2 = a_1 r\), \(a_3 = a_2 r = a_1 r^2\), me ētahi atu.
I te nuinga o te wā, ka taea te tuhi i te kupu \(n\) o tētahi raupapatanga ā-ira penei:
\[a_n = a_1 r^{(n-1)}\]
ko \(a_n\) te kupu \(n\) , ko \(a_1\) te kupu tuatahi, ā, ko \(r\) te ōwehenga.
Ngā Āhuatanga o te Raupapa Āhuahanga
1. Tauwehenga Pūmau:
He pumau tonu te ōwehenga i waenga i ngā kupu e rua e whai ake nei i roto i tētahi raupapatanga ā-ira. Mena ko te \(a_2 / a_1 = r\), ka noho tonu tēnei uara mō ngā takirua kupu katoa e whai ake nei.
2. Te Tipu Taupū:
Ka whakaatuhia e ngā raupapatanga ā-āhua me te ōwehenga \(r > 1\) te tipu taupū. I tetahi atu taha, ki te mea \(0 < r < 1\), ka whakaatu te raupapatanga i te pirau taupū.
3. Wāhanga Waenga: I roto i tētahi raupapatanga ā-ira, ko te wāhanga waenga o ngā wāhanga e toru e karapīpiti ana ko te toharite ā-ira o ngā wāhanga tuatahi me te tuatoru. Hei tauira, mēnā he toru ngā kupu karapīpiti a \(a, ar,\) me \(ar^2\), ko \(ar = \sqrt{a \cdot ar^2}\).
Ngā Whakamahinga o ngā Raupapa Āhuahanga E whakamahia ana ngā raupapa āhuahanga i roto i ngā mara maha nā te mea he āhuatanga taupū motuhake ō rātou. Anei ētahi tono nui: 1. Ōhanga me te Pūtea: I roto i ngā tataunga huamoni whakakotahi, ka tipu te moni kua whakangaohia i roto i te tauira raupapatanga ā-pūtaiao. Ki te haumi tetahi i te rupiah i te reiti huamoni o te r mō ia wā, ko te uara o te haumitanga i muri i ngā wā ko te P (1 + r)^n).
2. Ahupūngao: I roto i te ako i ngā wiri ā-hāngai me ngā ara iahiko hiko, ka whakamahia pinepinetia ngā raupapa ā-ira hei tātari i ngā kaha e heke ana, e piki ana rānei i ētahi wā.
3. Koiora: Ka taea e te taupori o ngā rauropi e whakaputa uri ana i roto i te taiao mutunga kore (tino pai) te tipu i runga anō i te ahunga ā-ira. Hei tauira, mā te tere tipu pumau, ka taea te tatau i te maha o ngā rauropi i roto i tētahi taupori mā te whakamahi i tētahi tātai mai i tētahi raupapatanga ā-ira.
Tauira Take 1. Tauira 1: Hoatu he raupapatanga me te kupu tuatahi \(a_1 = 3\) me te ōwehenga noa \(r = 2\). Nō reira, ka taea te tatau i te wāhanga tuarima o te raupapatanga mā te whakamahi i te tātai: \[a_5 = a_1 r^{(5-1)} = 3 2^4 = 3 16 = 48\] 2. Tauira 2: Me kī ka tāpui tētahi kaipupuri moni i te 1000 USD ki tētahi peeke me te 5% huamoni ia tau. E hia te moni ka toe i muri i ngā tau 10?
Ka taea te tatau i te uara whakamutunga o te haumitanga mā te: \[A = P (1 + r)^n\] ko \(P = 1000\), \(r = 0.05\), me \(n = 10\).
\[A = 1000 (1 + 0.05)^{10} = 1000 \cdot (1.05)^{10} = 1000 \cdot 1.62889 ≈ 1628.89\] Raupapa Āhuahanga Haunga ngā raupapa āhuahanga, kei reira anō te ariā o te raupapa āhuahanga, arā, ko te tapeke o ngā kupu i roto i te raupapa āhuahanga. Mena he raupapatanga ā-ira tā tātou \(a, ar, ar^2, \ldots, ar^{(n-1)}\), ka taea te tatau i te raupapa ā-ira tae atu ki te \(n\)th te kupu mā te whakamahi i te tātai: \[S_n = \frac{a (1 - r^n)}{1 - r} \; \text{for} \; r \neq 1\] Mō tētahi raupapa ā-ira mutunga kore me \(|r| <1\), ka tūtaki te tapeke o te raupapa, ā, ko te tātai koia tēnei: \[S = \frac{a}{1 - r}\] Tauira o tētahi Raupapa Ā-ira 1. Tauira 1: Raupapa Āhuahanga Mutunga Kore Hoatu he raupapa āhuahanga me te kupu tuatahi \(a = 4\), te ōwehenga noa \(r = 0.5\), ā, ka tāpirihia ki te kupu tuarima (\(n = 5\)). Kātahi, \[S_5 = \frac{4(1 - 0.5^5)}{1 - 0.5} = \frac{4(1 - 0.03125)}{0.5} = \frac{4 \cdot 0.96875}{0.5} = \frac{3.875}{0.5} = 7.75\] 2. Tauira 2: Raupapa Āhuahanga Mutunga Kore Mena he raupapa ā-irahanga tā tātou me \(a = 3\) me \(r = 1/3\), ko te tapeke o te raupapa mutunga kore ko: \[S = \frac{a}{1 - r} = \frac{3}{1 - \frac{1}{3}} = \frac{3}{\frac{2}{3}} = 3 \cdot \frac{3}{2} = \frac{9}{2} = 4.5\] Whakamutunga He taputapu kaha ngā raupapa ā-irahanga i roto i te pāngarau, me ngā tono mai i te ōhanga ki ngā pūtaiao taiao. Mā te mārama ki tēnei mea ka āwhina i te whakaoti rapanga maha e pā ana ki te tipu taupū, te pirau rānei. Mā te mōhio ki ngā ariā me ngā tātai o ngā raupapatanga ā-ira, ka taea e tātou te tātari me te mārama ki ngā āhuatanga rerekē o ia rā, o ia rā, o te ao mātauranga hoki.
