Ajụjụ Ihe Nlereanya Na-ekwu Maka Usoro Geometric
Usoro geometric bụ echiche dị oke mkpa na mgbakọ na mwepụ, nke na-apụtakarị n'ụdị nsogbu dị iche iche, gụnyere ule ụlọ akwụkwọ, ule ịbanye kọleji, na ọbụna ule ọkọlọtọ dịka SAT ma ọ bụ GRE. Nghọta zuru oke nke usoro geometric na-enyere anyị aka idozi nsogbu nke ọma. Isiokwu a ga-ekpuchi ọtụtụ ihe atụ nke nsogbu ma tụlee usoro geometric n'ụzọ zuru ezu.
Ịghọta Usoro Geometric
Usoro geometric bụ usoro ebe a na-enweta okwu ọ bụla site n'ịba ụba okwu gara aga site na ọnụọgụgụ edobere nke a na-akpọ ratio (ratio nkịtị, nke a na-ejikarị mkpụrụedemede \(r\) anọchite anya ya). N'ozuzu, enwere ike ide usoro geometric dị ka:
\[
a, ar, ar^2, ar^3, \ldots
\]
Ebe:
– \(a\) bụ okwu mbụ
– \(r\) bụ oke nke usoro ahụ
Ọ bụrụ na \( |r| < 1 \), usoro geometric na-enweghị ngwụcha nwere njirimara njikọta na-adọrọ mmasị. E nwere ọtụtụ ojiji bara uru nke usoro geometric n'ọtụtụ ngalaba dịka physics, akụnụba, na bayoloji. Usoro Usoro Geometric Okwu nke n'ime usoro geometric Enwere ike ịgbakọ okwu nke n'ime usoro geometric site na iji usoro: \[ U_n = a \cdot r^{n-1} \] Nchịkọta nke Usoro N'ime Usoro Geometric Enwere ike ịgbakọ mkpokọta nke okwu mbụ \(n\) nke usoro geometric (Sn) site na iji usoro: \[ S_n = a \frac{1 - r^n}{1 - r}, \quad \text{for } r \neq 1 \] \[ S_n = na, \quad \text{for } r = 1 \] Nchịkọta Na-enweghị Mbelata nke Usoro Geometric Ọ bụrụ na \(|r| < 1\), usoro geometric na-enweghị Mbelata nwere mkpokọta: \[ S_{\infty} = \frac{a}{1 - r} \] Ajụjụ na Mkparịta ụka Ihe atụ Ndị a bụ ụfọdụ ihe atụ nke ajụjụ usoro geometric yana mkparịta ụka ha: Ihe atụ Ajụjụ nke 1: Ịgbakọ Ajụjụ N'ime Okwu: E nyere usoro geometric nwere okwu mbụ \(a = 5\) na ratio \(r = 3\). Gbakọọ oge nke isii nke usoro ahụ. Mkparịta ụka: Iji usoro okwu nth: \[ U_6 = a \cdot r^{(6-1)} = 5 \cdot 3^5 = 5 \cdot 243 = 1215 \] Ya mere, okwu nke isii nke usoro ahụ bụ 1215. Ihe atụ Ajụjụ nke 2: Ịgbakọ Ọnụọgụ nke Okwu n Mbụ Ajụjụ: Gbakọọ mkpokọta nke okwu anọ mbụ nke usoro geometric na okwu mbụ \(a = 2\) na oke \(r = \frac{1}{2}\). Mkparịta ụka: Iji usoro maka mkpokọta okwu mbụ nke \(n\): \[ S_4 = a \frac{1 - r^4}{1 - r} = 2 \frac{1 - (\frac{1}{2})^4}{1 - \frac{1}{2}} = 2 \frac{1 - \frac{1}{16}}{\frac{1}{2}} = 2 \frac{\frac{15}{16}}{\frac{1}{2}} = 2 \cdot \frac{15}{8} = 2 \cdot \frac{15}{8} = \frac{30}{8} = 3.75 \] Ya mere, mkpokọta okwu anọ mbụ nke usoro ahụ bụ 3.75. Ihe atụ Ajụjụ nke 3: Mkpokọta Usoro Geometric Enweghị Mfe Ajụjụ: Gbakọọ mkpokọta usoro enweghị mfe ebe \(a = 7\) na \(r = \frac{1}{3}\). Mkparịta ụka: Iji usoro maka mkpokọta usoro enweghị ngwụcha: \[S_{\infty} = \frac{a}{1 - r} = \frac{7}{1 - \frac{1}{3}} = \frac{7}{\frac{2}{3}} = 7 \cdot \frac{3}{2} = \frac{21}{2} = 10.5 \] Ya mere, mkpokọta usoro enweghị ngwụcha bụ 10.5. Ihe atụ Ajụjụ nke 4: Ịchọpụta Usoro na Oke nke Ajụjụ Usoro: Ngụkọta nke usoro atọ mbụ nke usoro geometric bụ 21, na ngụkọta nke okwu nke abụọ na nke atọ bụ 18. Kpebie okwu mbụ na oke ya. Mkparịta ụka: Ka e were ya na okwu mbụ bụ \(a\) na oke ya bụ \(r\). Site na ozi nsogbu ahụ, anyị nwere ike ide nha nha abụọ ndị a: \[a + ar + ar^2 = 21 \quad \text{(1)} \] \[ar + ar^2 = 18 \quad \text{(2)} \] Site na nha nhata (2), anyị nwere ike ịkọwa \(a\) n'ihe gbasara \(r\): \[a(r + r^2) = 18 \itinye a = \frac{18}{r(1 + r)} \] Na-esote, dochie \(a\) n'ime nha nhata (1): \[ \frac{18(1)}{r(1 + r)} + \frac{18r}{r(1 + r)} + \frac{18r^2}{r(1 + r)} = 21 \] \[ \frac{18}{1 + r} + \frac{18r}{1 + r} + \frac{18r^2}{1 + r} = 21 \] \[ \frac{18 (1 + r + r^2)}{1 + r} = 21 \] \[ \frac{18 \cdot 3}{1 + r} = 21 \] \[ \frac{54}{1 + r} = 21 \] \[ 54 = 21(1 + r) \] \[ 54 = 21(1 + r) \] \[ 33 = 21r \] \[ r = \frac{33}{21} = \frac{11}{7} \] Mgbe a maara uru nke \(r\), dochie ya ọzọ na uru nke \(a\): \[ a = \frac{18}{r(1 + r)} = \frac{18}{\frac{11}{7} (21 + \frac{11}{7})} = \frac{18}{\frac{11}{7} \cdot \frac{18}{7}} = \frac{18 \cdot 7}{11 \cdot 18} = \frac{7}{11} \] Ya mere, okwu mbụ \(a\) bụ \(\frac{7}{11}\) na oke ya bụ \(\frac{11}{7}\). Mmechi Usoro Geometric bụ echiche mgbakọ na mwepụ nke a na-ejikarị eme ihe n'ọtụtụ ngwa. Ịghọta usoro dị mkpa dịka okwu n'ụzọ nke mbụ, mkpokọta nke okwu n'ụzọ nke mbụ, na mkpokọta nke usoro geometric na-enweghị ngwụcha dị oke mkpa maka idozi nsogbu mgbakọ na mwepụ dị iche iche metụtara ya. Site n'ịmụta ọtụtụ nsogbu ihe atụ dịka e kwuru n'isiokwu a, anyị nwere ike ime ka ikike anyị nwere ịghọta na iji usoro geometric ka mma.
