Ngā Tauira Pātai mō te Matapaki i ngā Raupapa Āhuahanga
He ariā taketake ngā raupapatanga ā-ira i roto i te pāngarau e whakaakona ana i te kura tuarua. Kei roto i ēnei raupapatanga he huinga tau, ko ia tau he hua o te tau o mua mā te pūmau e mōhiotia ana ko te "ōwehenga". Ka hipokina e tēnei tuhinga ētahi tauira rapanga me ngā matapakinga o ngā raupapatanga ā-ira, me te tumanako ka āwhina i ngā kaipānui ki te mārama ake ki tēnei ariā.
Te Mārama ki ngā Raupapa Āhuahanga
Ko te raupapatanga ā-ira he raupapatanga tau i hangaia mā te whakarea i te tau tuatahi (a) ki te ōwehenga pumau (r). I te nuinga o te wā, ko te āhua whānui o te raupapatanga ā-ira ko:
\[ a, ar, ar^2, ar^3, \ldots, ar^{n-1} \]
Anei:
– Ko “a” te kupu tuatahi o te raupapatanga.
– Ko te “r” te ōwehenga o tētahi wāhanga ki te wāhanga o mua.
– Ko “n” te kupu tuarima o te raupapatanga.
Ngā Pātai Tauira me te Kōrero
Me matapaki tātou i ētahi tauira rapanga kia nui ake ai tō māramatanga mō ngā raupapatanga ā-ira.
Tauira Pātai 1
Pātai:
Homai he raupapatanga ā-ira me te tau tuatahi (a) he 3, ā, ko te ōwehenga (r) he 2. Whakatauhia:
1. Te wāhanga 5 o te raupapatanga.
2. Te tapeke o ngā kupu tuatahi e 6 o te raupapatanga.
Kōrero:
1. Ka taea te tatau i te wāhanga tuarima (U5) mā te whakamahi i te tātai mō te wāhanga tuarima o tētahi raupapatanga ā-ira, arā:
\[ U_n = a \cdot r^{n-1} \]
Mā te whakakapi i a = 3, r = 2, me n = 5 ki roto i te tātai:
\[ U_5 = 3 \cdot 2^{5-1} \]
\[ U_5 = 3 \cdot 2^4 \]
\[ U_5 = 3 \cdot 16 \]
\[ U_5 = 48 \]
Nō reira, ko te 5 o ngā wāhanga o te raupapatanga ko te 48.
2. Ka taea te tatau i te tapeke o ngā kupu tuatahi e 6 (S6) o tētahi raupapatanga ā-ira mā te whakamahi i te tātai mō te tapeke o ngā kupu tuatahi n, arā:
\[ S_n = a \left( \frac{r^n – 1}{r – 1} \right) \]
Mā te whakakapi i a = 3, r = 2, me n = 6 ki roto i te tātai:
\[ S_6 = 3 \left( \frac{2^6 – 1}{2 – 1} \right) \]
\[ S_6 = 3 \left( \frac{64 – 1}{1} \right) \]
\[ S_6 = 3 \maui( 63 \matau) \]
\[ S_6 = 189 \]
Nō reira, ko te tapeke o ngā kupu tuatahi e 6 o te raupapatanga ko te 189.
Tauira Pātai 2
Pātai:
Ko te 27 te tau tuatoru o tētahi raupapatanga ā-ira, ā, ko te 243 te tau tuarima. Whakatauhia te uara o te tau tuatahi (a) me te ōwehenga (r).
Kōrero:
Homai a U3 = 27 me U5 = 243. Mā te whakamahi i te tātai mō te wāhanga tuarima o tētahi raupapatanga ā-ira:
\[ U_n = a \cdot r^{n-1} \]
Mō te U3:
\[ U_3 = a \cdot r^2 \]
\[ 27 = he \cdot r^2 \] \[ (1) \]
Mō te U5:
\[ U_5 = a \cdot r^4 \]
\[ 243 = he \cdot r^4 \] \[ (2) \]
Te whakatairite i ngā whārite (1) me (2) hei whakakore i te:
\[ \frac{U_5}{U_3} = \frac{a \cdot r^4}{a \cdot r^2} \]
\[ \frac{243}{27} = r^2 \]
\[ 9 = r^2 \]
\[ r = 3 \text{ or } r = -3 \]
Whakakapia te uara o r ki te whārite (1):
Mena ko te r = 3:
\[ 27 = he \cdot 3^2 \]
\[ 27 = a \cdot 9 \]
\[ a = 3 \]
Mena \( r = -3 \):
\[ 27 = he \cdot (-3)^2 \]
\[ 27 = a \cdot 9 \]
\[ a = 3 \]
Nō reira, ko te tau tuatahi (a) he 3, ā, ko te ōwehenga (r) ka taea te 3, te -3 rānei.
Tauira Pātai 3
Pātai:
Kimihia te tapeke mutunga kore o te raupapa ā-ira e whai ake nei mēnā ko te 8 te wāhanga tuatahi (a) ā, ko te 1/2 te ōwehenga (r).
Kōrero:
Ka taea te tatau i te tapeke mutunga kore o tētahi raupapa āhuahanga mā te whakamahi i te tātai:
\[ S_{\infty} = \frac{a}{1 – r} \]
Mā te whakakapi i a = 8 me r = 1/2 ki te tātai:
\[ S_{\infty} = \frac{8}{1 – \frac{1}{2}} \]
\[ S_{\infty} = \frac{8}{\frac{1}{2}} \]
\[ S_{\infty} = 8 \times 2 \]
\[ S_{\infty} = 16 \]
Nō reira, ko te tapeke mutunga kore o te raupapa āhuahanga ko te 16.
Tauira Pātai 4
Pātai:
Ko te 12 te tau tuarua o tētahi raupapatanga ā-ira, ā, ko te 108 te tau tuawhā. Whakatauhia te ōwehenga me te tau tuatahi o te raupapatanga.
Kōrero:
Homai \( U_2 = 12 \) me \( U_4 = 108 \). Mā te whakamahi i te tātai mō te wāhanga tuarima o tētahi raupapatanga ā-ira:
Mō \( U_2 \):
\[ U_2 = a \cdot r \]
\[ 12 = a \cdot r \] \[ (1) \]
Mō \( U_4 \):
\[ U_4 = a \cdot r^3 \]
\[ 108 = he \cdot r^3 \] \[ (2) \]
Te whakatairite i ngā whārite (1) me (2) hei whakakore i te:
\[ \frac{U_4}{U_2} = \frac{a \cdot r^3}{a \cdot r} \]
\[ \frac{108}{12} = r^2 \]
\[ 9 = r^2 \]
\[ r = 3 \text{ or } r = -3 \]
Whakakapia te uara o r ki te whārite (1):
Mena ko te r = 3:
\[ 12 = a \cdot 3 \]
\[ a = 4 \]
Mena \( r = -3 \):
\[ 12 = a \cdot (-3) \]
\[ a = -4 \]
Nō reira, ko te tau tuatahi (a) ka taea te 4, te -4 rānei, ā, ko te ōwehenga (r) ka taea te 3, te -3 rānei.
Whakamutunga
He ariā pāngarau nui ngā raupapatanga ā-ira e whakamahia whānuitia ana i roto i ngā momo mara. Mā te mārama ki ngā kaupapa matua me te whakaharatau i ngā pūkenga whakaoti rapanga, ko te tumanako ka taea e tātou te matatau ake ki te mārama me te whakamahi i te ariā. Kei roto i tēnei tuhinga ētahi tauira rapanga me ngā kōrero hei āwhina i ngā kaipānui ki te ako me te mārama hohonu ake ki ngā raupapatanga ā-ira. Ko te tumanako ka whai hua tēnei!
