Ngā Tauira Pātai e Matapaki ana i ngā Raupapa me ngā Raupapa
He ariā taketake ngā raupapatanga me ngā raupapatanga i roto i te pāngarau, e kitea pinepinetia ana mai i te kura tuatahi ki te whare wānanga. Ko te raupapatanga he kohinga tau kua whakaritea kia rite ki tētahi ture motuhake, ko te raupapatanga ia ko te tapeke o ngā kupu o taua raupapatanga. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira raruraru me ngā raupapatanga me ngā raupapatanga.
Tauira 1: Raupapa Pāngarau
Pātai:
Homai he raupapatanga tātai me te kupu tuatahi (a) = 3 me te rerekētanga (d) = 5. Whakatauhia:
1. Te wāhanga 10 o te raupapatanga.
2. Te tapeke o ngā kupu tuatahi e 20 o te raupapatanga.
Kōrero:
1. Wāhanga 10
Ko te tātai mō te wāhanga tuarima o tētahi raupapatanga pāngarau ko:
\[
U_n = a + (n-1)d
\]
Mō te wāhanga 10 (U_10):
\[
U_{10} = 3 + (10-1) \cdot 5 = 3 + 45 = 48
\]
2. Te Tapeke o ngā Kupu Tuatahi e 20
Ko te tātai mō te tapeke o ngā kupu tuatahi n (S_n) o tētahi raupapatanga taunga ko:
\[
S_n = \frac{n}{2} (2a + (n-1)d)
\]
Mō te tapeke o ngā kupu tuatahi e 20 (S_20):
\[
S_{20} = \frac{20}{2} (2 \cdot 3 + (20-1) \cdot 5) = 10 (6 + 95) = 10 \cdot 101 = 1010
\]
Tauira 2: Raupapa Āhuahanga
Pātai:
Hoatu he raupapatanga ā-ira me te tau tuatahi (a) = 4 me te ōwehenga (r) = 2. Whakatauhia:
1. Te wāhanga 6 o te raupapatanga.
2. Te tapeke o ngā kupu tuatahi e 8 o te raupapatanga.
Kōrero:
1. Wāhanga 6
Ko te tātai mō te tau tuarima o tētahi raupapatanga ā-ira ko:
\[
U_n = a \cdot r^{(n-1)}
\]
Mō te wāhanga 6 (U_6):
\[
U_{6} = 4 \cdot 2^{(6-1)} = 4 \cdot 2^5 = 4 \cdot 32 = 128
\]
2. Te Tapeke o ngā Kupu Tuatahi e 8
Ko te tātai mō te tapeke o ngā kupu tuatahi n (S_n) o tētahi raupapatanga ā-ira ko:
\[
S_n = a \frac{r^n – 1}{r – 1}
\]
Mō te tapeke o ngā kupu tuatahi e 8 (S_8):
\[
S_{8} = 4 \frac{2^8 – 1}{2 – 1} = 4 \frac{256 – 1}{1} = 4 \cdot 255 = 1020
\]
Tauira 3: Raupapa Mutunga Kore Āhuahanga Whakakotahi
Pātai:
Hoatu he raupapa ā-ira me te kupu tuatahi (a) = 1 me te ōwehenga (r) = 1/2. Whakatauhia te tapeke o te raupapa mutunga kore.
Kōrero:
Ko te tātai mō te tapeke o tētahi raupapa mutunga kore (S_∞) o tētahi raupapa āhuahanga taupatupatu ko:
\[
S_{\infty} = \frac{a}{1 – r}
\]
Nā, mō tēnei raupapa:
\[
S_{\infty} = \frac{1}{1 – \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2
\]
Tauira 4: Ngā Raupapa me ngā Raupapa o ngā Tau Tapawhā
Pātai:
Mēnā ka hoatu he raupapatanga o ngā tau tapawhā me te wāhanga tuatahi (U_1) = 1, te wāhanga tuarua (U_2) = 4, me te wāhanga tuatoru (U_3) = 9. Whakatauhia te wāhanga tuarima o te raupapatanga. He raupapatanga tātai, he raupapatanga ā-ira rānei tēnei raupapatanga? Whakamāramahia.
Kōrero:
1. Wāhanga 5
Ko te tauira o tētahi raupapatanga o ngā tau tapawhā ko:
\[
U_n = n^2
\]
Mō te wāhanga 5 (U_5):
\[
U_{5} = 5^2 = 25
\]
2. Momo Raina
Hei tirotiro mēnā he raupapatanga ā-tau, ā-ira rānei tēnei, tirohia te rerekētanga i waenga i ngā kupu (te rerekētanga noa) me te ōwehenga i waenga i ngā kupu:
– Te rerekētanga i waenga i ngā kupu (d):
\[
U_2 – U_1 = 4 – 1 = 3
U_3 – U_2 = 9 – 4 = 5
\]
Nā te mea kāore te rerekētanga i te pumau, ehara tēnei raupapatanga i te pāngarau.
– Te ōwehenga i waenga i ngā iwi (r):
\[
\frac{U_2}{U_1} = \frac{4}{1} = 4 \\
\frac{U_3}{U_2} = \frac{9}{4} = 2.25
\]
Nā te mea kāore te ōwehenga i waenga i ngā kupu i te pūmau, ehara tēnei raupapatanga i te āhua ā-ira.
Nō reira, ehara tēnei raupapatanga o ngā tau tapawhā i te tātaitanga, ehara hoki i te raupapatanga ā-ira, engari he raupapatanga motuhake e whai ana i te tauira o ngā tau tapawhā.
Tauira 5: Raupapa Pāngarau Mutunga Kore
Pātai:
Ka taea te tatau i te tapeke o tētahi raupapa pāngarau mutunga kore? Mena koinā, homai he tauira. Ki te kore, whakamāramahia he aha.
Kōrero:
Kāore i rite ki ngā raupapa ā-ira, ko ngā raupapa tātai mutunga kore he tapeke mutunga kore. Nā te mea ka piki, ka heke rānei ia wāhanga i runga i te rārangi, nō reira ka tipu tonu te tapeke mō ake tonu atu.
Hei tauira, whakaarohia he raupapa pāngarau mutunga kore me te wāhanga tuatahi 1 me te rerekētanga noa 1:
\[
1 + 2 + 3 + 4 + \ldots
\]
Ki te ngana tātou ki te tāpiri i a rātou, ka mārama kāore te raupapa e tūtaki ki tētahi uara pumau, engari ka tata ki te mutunga kore. Nō reira, ko te tapeke o tētahi raupapa tātai mutunga kore, i te nuinga o te wā, he mutunga kore, ā, kāore e taea te tatau pērā i tētahi raupapa āhuahanga tūtaki.
-
I roto i tēnei tuhinga, kua hipokina e mātou ētahi tauira rapanga, ā, kua kōrerohia hoki ngā raupapatanga me ngā raupapatanga. Kua arotakehia e mātou ngā raupapatanga tātai me ngā raupapatanga ā-ira, kua kite i te huarahi ki te tatau i te wāhanga tuarima me te tapeke o ō rātou wāhanga tuatahi, ā, kua whakautua hoki ngā pātai mō ngā raupapatanga mutunga kore. Mā te mārama ki ēnei ariā me ngā tauira taketake, ko te tumanako ka nui ake tō māia i a koe e whakatata atu ana ki ngā raupapatanga me ngā raupapatanga i roto i te pāngarau.
