Geometric kunoita. MUENZANISO pachisarudzo

Funga pamusara mumwe.

7 28 112 448 1792 ...

chinoratidza kwazvo zvakajeka kuti kukosha chero zvinhu zvayo kupfuura yapfuura chaizvo mana nguva. Saka, nyaya iri kunoita.

geometrical kunoita akadana magumo kutevedzana nhamba, chikuru aizoonekawo kuti nhamba inotevera inotengwa kubva pamusoro kuburikidza kuwedzera zvimwe kwenhamba. Izvi kunoratidzwa zvinotevera payakavakirwa.

_{Z +1 = Z · Q} , apo z - nhamba yasarudzwa maatomu akafanana.

Saizvozvowo, z ∈ N.

A nguva apo chikoro anodzidzwa geometric kunoita - giredhi 9th. Mienzaniso kuchakubatsira kunzwisisa pfungwa:

0.25 0.125 0.0625 ...

18 February 6 ...

Kubva musi yemaitiro ichi, kuchiita kuti dhinomineta kuwanikwe sezvinotevera:

Kana Q, kana b _z havagoni kuva razero. Uyewo, mumwe nomumwe ne munhevedzano nhamba kunoita hakufaniri razero.

Saka, kuona inotevera nhamba nhamba, muwande yokupedzisira nokuda q.

Chidzore kunoita ichi, unofanira kutaridzwa chinhu chacho uye dhinomineta wokutanga. Pashure paizvozvo zvinokwanisika kuwana chero zvinotevera nhengo dzavo uwandu.

mhando

Zvichienderana Q uye _1, kunoita ichi rakakamurwa marudzi akawanda:

• Kana _1, uye Q mukuru mumwe, ndokubva kutevedzana - kuwedzera mumwe anotevedzana che munhu geometric kunoita. Mienzaniso yaro hunonyatsodonongodza pazasi.

Muenzaniso: ₁ = 3, q = 2 - kupfuura kubatana, vose parameters.

Zvadaro kutevedzana kwezviitiko nhamba anogona kunyorwa se:

3 6 12 24 48 ...

• Kana | q | asingasviki rimwe, kureva, zviri yakaenzana chokuwanziridza kubudikidza nekupesana, kuchinja zvakafanana ezvinhu - kudzikira geometric kunoita. Mienzaniso yaro hunonyatsodonongodza pazasi.

Muenzaniso: ₁ = 6, q = 1/3 - ₁ mukuru mumwe, Q - zvishoma.

Zvadaro kutevedzana kwezviitiko nhamba anogona kunyorwa seizvi:

June 2 2/3 ... - chero chinhu yakawanda zvinhu zvinotevera nayo, ndicho 3 nguva.

• Rakatumbuka. Kana Q <0, zviratidzo kuwanda kutevedzana rakatumbuka nguva dzose zvisinei ane _1, uye zvinhu zvinoumba yemichero kana kuderera.

Muenzaniso: ₁ = -3, q = -2 - zvose asingasviki razero.

Zvadaro kutevedzana kwezviitiko nhamba anogona kunyorwa se:

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

fomura

For yakakodzera kushandiswa, pane zvakawanda geometric progressions of Masvomhu:

• Formula Z-fung izwi. Kunoitawo dzemaitirwe ari chinhu imwe nhamba pasina pakuverenga yapfuura nhamba.

Muenzaniso: q = 3, mumwe = ₁ 4. kunodiwa kuverenga chimwe chinhu chechina kunoita.

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

• Verengai okutanga okunze, ane nhamba akaenzana Z. Kunoitawo dzemaitirwe pamusoro Verengai zvinhu zvose kutevedzana kuti _Z zvichibatanidzwa.

≠ 0, saka, Q haasi 1 - (Q 1) Sezvo (1- q) ari dhinomineta, ipapo.

Cherechedza: kana q = 1, ipapo kufambira mberi ringadai anomiririrwa akawanda nokusingaperi kudzokorora nhamba.

Amount exponentially mienzaniso: ₁ = 2, q = -2. Change S _5.

Solution: S ₅ = 22 - kukarukureta fomura.

• Amount kana | q | <1 uye kana z ruchigona kusaverengeka.

Muenzaniso: ₁ = _2, q = 0.5. Find Verengai.

Solution: S _z = 2 x = 4

Kana isu ngaaverenge Verengai anoverengeka nhengo manual, uchaona kuti zvechokwadi kuita kuti vana.

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

Vamwe ehupfumi:

• A mamirire pfuma. Kana zvinotevera ezvinhu Chine upi Z, ipapo akapiwa kusanganiswa akatevedzana - munhu geometric kunoita:

_Z ² = _{Z -1} · _{Z + 1}

• Ndirwowo mativi upi nhamba iri exponentially achishandisa Kuwedzerwa dzinoungana vemamwe maviri nhamba chero musara wakapiwa, kana vari equidistant kubva maatomu akafanana.

² _Z = _{Z - t} ² _{+ Z} + _t ² apo T - daro pakati nhamba idzi.

• vanosiyana Q nguva okunze.
• The logarithms of zvinhu kunoita pamwe kuumba kunoita, asi svomhu, kureva, mumwe nomumwe wavo kupfuura rapfuura mumwe imwe nhamba.

Mienzaniso zvimwe zvinetso dzenhoroondo

Kuti tinzwisise zviri nani zvinotaurwa geometric kunoita, nechisarudzo mienzaniso kugiredhi 9 kunogona kubatsira.

• Terms uye mamiriro: ₁ = 3, ₃ = 48. Find q.

Solution: mumwe anotevedzana chinhu kupfuura yapfuura Q nguva. Zvakakosha kuratidza zvimwe zvinhu kuburikidza dzimwe Via dhinomineta.

Somugumisiro, mumwe ₃ = Q ² · ₁

Kana dzinomiririra q = 4

• Conditions: ₂ = 6, mumwe = ₃ 12. Change S _6.

Solution: Kuti kuita izvi, zvakatikwanira kuwana Q, wokutanga maatomu akafanana uye chinotsiva kupinda fomura.

₃ = Q · _2, naizvozvo, q = 2

₂ = Q · A _1, saka mumwe = ₁ 3

S = ₆ 189

• · A ₁ = 10, q = -2. Find wechina vanzwe kunoita.

Solution Zvaringana kuti ataure chinhu chechina kuburikidza wokutanga uye kubudikidza dhinomineta.

₄ ³ = Q · munhu = ₁ -80

Application muenzaniso:

• Bank yeGmail.END_STRONG chakakonzera Verengai 10,000 rubles, pasi iro gore nemhosva kune chikuru chitsama zvichawedzerwa 6% payo zvazvo. How mari yakawanda iri nhoroondo pashure 4 Makore?

Solution: The pakutanga mari yakaenzana zviuru 10 rubles. Saka, gore pashure upfumi munyaya chichava mari akaenzana 10000 + 10000 = 10000 · 0,06 · 1,06

Nokudaro, uwandu nhoroondo kunyange pashure gore rimwe kuitwa sezvinotevera:

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

Kureva, rimwe gore mari yakawedzera kusvika 1,06 nguva. Saka, kuti kuwana nhamba nhoroondo pashure 4 makore zvakatikwanira kuwana chinhu chechina kunoita, unopirwa wokutanga maatomu akafanana akaenzana 10 zviuru, uye dhinomineta akaenzana 1,06.

S = 1,06 · 1,06 · 1,06 · 1,06 · 10000 = 12625

Mienzaniso zvinetso computation iri kuwanda:

In matambudziko akasiyana-siyana achishandisa geometric kunoita. Muenzaniso yokuwana Verengai anogona kuisa sezvinotevera:

₁ = 4, q = 2, overenga S _5.

Solution: ose anodiwa Data kuti kuverenga vanozivikanwa, anongoti kutsiva muvira fomura.

S ₅ = 124

• ₂ = 6, mumwe = ₃ 18. Change Verengai matanhatu zvinhu vokutanga.

mhinduro:

The Geom. kufambira mberi mumwe chinhu rakatevera yakakura kupfuura rapfuura Q nguva, kureva kuti kuverenga mari unofanira kuziva chinhu ₁ uye dhinomineta q.

₂ · q = ₃

q = 3

Saizvozvowo, tinofanira kuwana _{1, 2} uye kuziva q.

₁ · q = ₂

₁ = 2

Zvino zvakatikwanira kuti kutsiva known mashoko kupinda fomura uwandu.

S ₆ = 728.