Kulandelana Geometric. UMZEKELO kwisigqibo

Cinga emgceni.

7 28 112 448 1792 ...

ubonisa Kucace gca ukuba ixabiso lazo naziphi na izinto zayo ngaphezu lwangaphambili kanye kane. Ngoko ke, olu ngcelele a ukuqhubela phambili.

zejometri phambili ngokuba ulandelelwano engenasiphelo lwamanani, iphambili leyo kukuba le nombolo ilandelayo lufumaneka ngasentla yokuphindaphinda nge inombolo ethile. Oku kubonakaliswa indlela elandelayo.

_{a z +1 = a z · q} , apho z - inombolo element ekhethiweyo.

Ngako oko, z ∈ N.

A ixesha xa isikolo olufundwa zejometri progression - grade 9. Imizekelo iya kukunceda uqonde magama:

0.25 0.125 0.0625 ...

18 Februwari 6 ...

Ngokusekelwe kule formula, ukuqhubela phambili elingezantsi zingafumaneka ngolu hlobo lulandelayo:

Kanjalo _z q, okanye b ayikwazi ukuba uziro. Kwakhona, ngasinye kuzo iziqalelo uthotho amanani kufuneka ibe progression zero.

Ngako oko, ukuze ubone inani elilandelayo lwenani, wabandisa yokugqibela yi q.

Ukuchaza lo progression, kufuneka ukhankanye i element wokuqala kuyo kunye nenani eliphantsi. Emva ukuba kuyenzeka ukufumana nayiphi na malungu alandelayo kunye nemali yabo.

uhlobo

Kuxhomekeka q kunye _1, le yenkqubela zohlulwa ngokwendidi ezininzi:

• Ukuba _1, kunye q mkhulu kunalowo, ngoko ulandelelwano - elandayo kunye isiqalelo ngasinye ezilandelelanayo a progression yemigca. Imizekelo yalo zi ngezantsi.

Umzekelo: a ₁ = 3, q = 2 - mkhulu umanyano, zombini parameters.

Emva koko ukulandelelana kwamanani angabhalwa ngolu hlobo:

3 6 12 24 48 ...

• Ukuba | q | ngaphantsi kwe enye, oko kukuthi, oku kulingana-phinda ngo-hlula, ukuqhubela phambili kunye neemeko ezifanayo - liyehla zejometri zokuqhubela phambili. Imizekelo yalo zi ngezantsi.

Umzekelo: a ₁ = 6, q = 1/3 - ₁ mkhulu kunalowo, q - ngaphantsi.

Emva koko ukulandelelana kwamanani angabhalwa ngolu hlobo lulandelayo:

6 2 2/3 ... - izinto elementi xaxa oko, amaxesha-3.

• Nokuxuba. Ukuba q <0, imiqondiso amanani ifildi ngokulandelelana rhoqo nokuba _a-1, kwaye izinto zayo nayiphi ukwanda okanye ukuncipha.

Umzekelo: _i-1 = -3, q = -2 - zombini libengaphantsi kunonothi.

Emva koko ukulandelelana kwamanani angabhalwa ngolu hlobo:

3, 6, -12, 24, ...

ifomula

Ukusetyenziswa lula, kukho progressions ezininzi zejiyometri le fomyula:

• Formula z-th elide. Ivumela ukubalwa sakhi inani elithile ngaphandle yokubala amanani edlulileyo.

Umzekelo: q = 3, a = ₁ 4. ezifunekayo ukubala uchatha into yesine.

Isisombululo: a = ₄ 4 3 · ^4-1 · 3 = 4 ³ = 4 · 27 = 108.

• Umdibaniso izinto zokuqala, ogama inani uyalingana z. Ivumela ukubalwa isixa zonke izinto ngolandelelwano ukuya ebandakanya wonke _z.

≠ 0, ngaloo ndlela, q ayikho 1 - (q 1) Ekubeni (1- q) kwinani elingaphezulu, ngoko.

Qaphela: Ukuba q = 1, ngoko ke progression ange emelwe inani ngonaphakade ngokuphinda inani.

Imali lunyuka imizekelo: ₁ = 2, q = -2. Bala S _5.

Isisombululo: S ₅ = 22 - calculation ifomula.

• Isixa xa | q | <1 xa z kuyaphela yokungapheli.

Umzekelo: a ₁ = _2, q = 0.5. Fumana inani.

Isisombululo: S _z = 2 x = 4

Ukuba Ungabala udibaniso amalungu eKhabhinethi manual, uya kubona ukuba eneneni uzinikele ezine.

S _z = + 1 + 2 0.5 + 0.25 + 0,125 + 0.0625 = 3.9375 4

Ezinye iimpawu:

• A impahla isici. Ukuba le meko zilandelayo Ibeka nasiphi z, koko kunikwa uludwe lwamanani - umkhuba lwezibalo:

a _z ² = A _{z -1} · A _{z + 1}

• Kubalulekile kwakhona isikweri naliphi na inani ngokucacileyo ngokusebenzisa Ukongeza lwezikweri amanye amanani amabini nawuphi na umqolo lona, ukuba equidistant ukusuka element.

² a _z = a _{z - t} ² ₊ a _z + _t ² apho t - umgama phakathi kwala manani.

• Iziqalelo abafani amaxesha q.
• I see logarithm ye amasuntswana progression ngendlela kwakunye ukuqhubela phambili, kodwa arithmetic, oko kukuthi, ngamnye kubo ngaphezu ezidlulileyo ngokwenani.

Imizekelo ezinye iingxaki yamandulo

Ukuze siqonde ngcono ukuba yintoni ukuqhubela zejometri, kunye imizekelo isigqibo kwibakala 9 kunokunceda.

• Imigaqo nemiqathango: ₁ = 3, ₃ = 48. Fumana q.

Isisombululo: isiqalelo ngasinye wempumelelo ngo ngaphezu q yangaphambili ixesha. Kuyimfuneko ukuba ukuchaza izinto ezithile ngokusebenzisa enye nge kwinani elingezantsi.

Ngenxa yoko, _i-3 = q ² · ₁

Xa kufakwe q = 4

• Imiqathango: ₂ = 6, a = ₃ 12. Bala S _6.

Isisombululo: Ukuze wenze oku, lwanele ukufumana q, isiqalelo yokuqala kunye ababambeleyo kungena ifomula.

₃ = q · _a-2, ngenxa yoko, q = 2

a ₂ = q · A _1, ngoko a = ₁ 3

S = ₆ 189

• · A ₁ = 10, q = -2. Fumana element wesine ukuqhubela phambili.

Isisombululo: kwanele ukubonisa i element wesine ngokubanika wokuqala yaye elingezantsi.

₄ ³ = q · a = ₁ -80

Umzekelo yesicelo:

• iklayenti Bank iye negalelo isixa engange 10,000, phantsi apho nyaka ngamnye umxhamli isixa inqununu iya kongezwa-6% kwayo kodwa. Ingakanani imali kwi-akhawunti emva kweminyaka-4?

Isisombululo: yokuqala mali esilingana amawaka 10 engange. Ngoko ke, unyaka emva kokuba notyalo-akhawunti iya kuba isixa esilingana 10000 + 10000 = 10000 · 0,06 · 1.06

Ngako oko, le mali kwi-akhawunti emva konyaka omnye uza kulubonakalisa ngolu hlobo lulandelayo:

(10000 · 1.06) · 10000 · 0,06 + 1,06 = 1,06 · 1.06 · 10000

Oko kukuthi, ngonyaka ngamnye imali olwandisiweyo amaxesha 1.06. Ngenxa yoko, ukuba ukufumana inani akhawunti emva kweminyaka 4, lwanele ukufumana uchatha yesine element, onikelwa Into yokuqala elingana 10 lamawaka, kunye nenani elilinganayo 1.06.

S = 1.06 · 1.06 · 1.06 · 1.06 · 10000 = 12625

Imizekelo yeengxaki xa kubalwa i wamcacisela inani:

Xa iingxaki ezahlukeneyo besebenzisa zejometri zokuqhubela phambili. Umzekelo lokufumana isixa lungasetelwa ngolu hlobo lulandelayo:

₁ = 4, q = 2, ukubala S _5.

Isisombululo: zonke iinkcukacha eziyimfuneko ukuze ukubala baziwa, ababambeleyo nje ebakhokelela ifomula.

S ₅ = 124

• ₂ = 6, a = ₃ 18. Bala inani iinkalo ezintandathu zokuqala.

isicombululo:

I Geom. inkqubela isiqalelo ngasinye elikhulu elandelayo ngaphezu amaxesha q odlulileyo, oko kukuthi, ukubala imali kufuneka wazi element ₁ kunye q eliphezulu nenani eliphantsi.

a ₂ · q = ₃

q = 3

Ngokufanayo, imfuneko ukufumana _{1, 2} kunye abazi q.

a ₁ · q = ₂

₁ = 2

Kwaye ngoko lwanele kufakwe idatha ezaziwayo kwi mali ifomula kulo.

S ₆ = 728.