Geometric kunoita chacho zvivako

Geometric kunoita kunokosha mune masvomhu somunhu sainzi, uye vakashandisa ukoshi, sezvo ane zvikuru yakafara zvezvinotaurwa, kunyange yepamusoro masvomhu, somuenzaniso, ari dzidziso akatevedzana. mashoko okutanga mberi vakauya kwatiri kubva muIjipiti yekare, kunyanya muchimiro dambudziko inozivikanwa of Rhind nenhokwe nomwe vanhu vane katsi nomwe. Misiyano basa iri vakanga kudzokororwa kakawanda panguva dzakasiyana nemamwe marudzi. Kunyange Velikiy Leonardo Pizansky, rinozivikanwa Fibonacci (XIII c.), Akataura naye ari ake "Book of Mushosho."

Zvekuti geometric kunoita aine kare. Rinomiririra kusanganiswa kutevedzana ane nonzero wokutanga nhengo, uye mumwe nomumwe hunotevedzana, kutanga wechipiri vagara vandifungira vachiwedzera rapfuura dzokororo payakavakirwa panguva dzose, nonzero nhamba inonzi dhinomineta kunoita (zvinowanzoda akasarudzwa vachishandisa tsamba q).
Zviri pachena, rinogona kuwanikwa kupatsanura mumwe Zvakazoitwa rokuti kutevedzana kusvikira yapfuura, i.e. z 2: z 1 = ... = zn: z N-1 = .... Somugumisiro, vakawanda basa kunoita (zn) zvakakwana kuti anoziva kukosha rokutanga rokuti the dhinomineta uye and 1 q.

Somuenzaniso, regai Z 1 = 7, q = - 4 (Q <0), ipapo zvinotevera geometric kunoita inotengwa 7 - 28, 112 - 448, .... Sezvaunogona kuona, kunoguma kutevedzana haisi monotone.

Yeuka kuti imwe zvemasanga kutevedzana kwezviitiko monotonous (kuwedzera / kudzikira) apo imwe nhengo yayo vanotevera zvakawanda / zvishoma pane yapfuura mumwe. Somuenzaniso, kutevedzana 2, 5, 9, ..., uye -10, -100, -1000, ... - Monotone, wechipiri chete - a kudzikira geometric kunoita.

Kana apo q = 1, mitezo yose zvinowanikwa kuti, uye anonzi dzose kunoita.

Kutevedzana raiva kuchiita ichi mhando, inofanira kugutsa zvinotevera zvakafanira uye zvakakwana ezvinhu, zvinoti: kubvira wechipiri, neimwe nhengo dzayo dzinofanira kuva geometric zvinoreva kuti nawo nhengo.

pfuma iyi inobvumira pasi vamwe vaviri pedyo kuwanikwa zvemasanga rokuti kunoita.

N-fung izwi exponentially nyore kuwanikwa ne nzira: zn = z 1 * Q ^ (n-1), z muchiziva pakutanga nhengo 1 uye dhinomineta q.

Sezvo nhamba kutevedzana ane chitsama, ipapo zvishomanana Masvomhu kutipa nzira kuverenga kuwanda kuchiita nhengo, anoti pakutanga:

S n = - (zn * Q - Z 1) / (1 - q).

Pavanotsiva, ari nzira kwayo okuti ukoshi zn z 1 * Q ^ (n-1) kuwana kechipiri Nemutengo yokuita kuchinja: S n = - Z1 * (Q ^ N - 1) / (1 - q).

Akakodzera kuteererwa inotevera dzinonakidza chokwadi: ivhu hwendefa inowanikwa Zvakafukunurwa Bhabhironi rekare, iro rinoreva VI. BC, rine nzira inoshamisa Verengai 1 + 2 + ... + 22 + 29 akaenzana 2 wegumi simba kubvisa 1. The tsananguro Fani iyi haisati yawanikwa.

Tinoona mumwe zvivako zviri geometric kunoita - kuramba basa nhengo dzayo, munzvimbo dzokuzororera dzaiva pa zvakaenzana kure kumagumo kutevedzana.

Of kunyanya kukosha kubva esayenzi maonero, zvakadai chinhu sezvo isingaperi geometric kunoita uye pakuverenga uwandu kwayo. Vachifunga kuti (Yn) - munhu geometric kunoita ane dhinomineta Q, kugutsa ezvinhu | q | <1, uwandu ayo achaitwa raireva pokupedzisira kune yatiri munotoziva Verengai nhengo dzaro wokutanga, pamwe kunofashukira nokuwedzera N, saka nazvo kusvika kusaverengeka.

Find mari iyi somugumisiro kushandisa nzira:

S N = y 1 / (1- q).

Uye, sezvo ruzivo akaratidza, nokuti pachena nyore kunoita ichi chakavanzika chihombe kushanda nekugona. Somuenzaniso, kana tiri kuvaka kutevedzana kwezviitiko dzinoungana maererano zvinotevera algorithm, unobva midpoints ari yaimbovapo, ipapo zvinoumba mativi parinogumira geometric kunoita ane dhinomineta 1/2. Chete kunoita chimiro uye nzvimbo zvokuridza, vakawana padanho rimwe nerimwe kuvakwa, uye ayo makuru akaenzana nzvimbo yepakutanga dzomumusha.