Horumarka joomateri iyo hantida

horumarka joomateri waxaa muhiim ah in xisaabta sida sayniska, iyo codsatay muhiimadda, tan iyo markii ay isugu dhafan aad u ballaadhan, taasoo ah xisaabta sare, tusaale ahaan, in aragtida ah ee taxanahan. Macluumaadka ugu horeysay horumarka nooga yimid qadiimiga Masar, gaar ahaan qaab dhibaato si fiican u yaqaan caw Rhind toddobada qof toddoba bisadaha. Isbadalka ee hawshan ayaa soo noqnoqda marar badan waqtiyo kala duwan ka soo dhacay quruumihii kale. Xitaa Leonardo Velikiy ku Pizansky, oo loo yaqaan Fibonacci (XIII c.), La hadleen ayay ku in uu "Book of Abacus ah."

Sidaas daraaddeed in horumarka joomateri leedahay taariikh guun ah. Waxay ka dhigan tahay isku xigxiga oo tiro la xubin nonzero ugu horeysay, iyo mid kasta oo xiga, laga bilaabo labaad waxaa lagu go'aamiyaa tarmo caanaha bogsashada hore tiro joogto ah, nonzero la yidhaahdo horumarka hooseeyaha (waxaa sida caadiga ah loogu talagalay adiga oo isticmaalaya warqad q ah).
Sida iska cad, waxaa laga heli karaa qeybi xilli kasta xiga ee isku xigxiga si hore, i.e. z 2: z = 1 ... = amayno: z n-1 = .... Sidaas awgeed, inta badan horumarka shaqada (amayno) ku filan in ay og yahay qiimaha muddo xileedka koowaad ee hooseeyaha iyo y 1 q.

Tusaale ahaan, ha z 1 = 7, q = - 4 (q <0), ka dibna soo socda horumarka joomateri waa la helay 7 - 28, 112 - 448, .... Sida aad arki karto, isku xigxiga ka dhalanaya ma aha monotone.

Xusuustaa xigxiga ah aan loo aabo yeelin ee sugeysayna (kordhaya / hoos) markii mid ka mid ah xubnaha raaci more / ka yar hal hore. Tusaale ahaan, u kala horreeyaan 2, 5, 9, ..., iyo -10, -100, -1000, ... - Monotone, labaad mid ka mid - a hoos horumarka joomateri.

In kiiska halkaas oo q = 1, xubnaha oo dhammu waxaa la helay in ay noqon, iyo waxa la yidhaahdaa waa horumarka joogto ah.

taxanaha ahaa horumarka noocan ah, waa in ay ka dhergin xaalad soo socda loo baahan yahay oo ku filan, kuwaas oo: laga bilaabo labaad, mid kasta oo xubnaha waa in ay ahaataa celceliska joomateri xubnaha la deriska ah.

hantida Tani waxay u ogolaaneysaa hoos baa laba raadinta xiga horumarka muddada loo aabo yeelin.

muddo n-th aheyn si fudud looga heli by formula ah: amayno = z 1 * q ^ (n-1), z og xubin marka hore 1 iyo hooseeyaha q ah.

Tan iyo markii ay kala horreeyaan tirada uu leeyahay lacag, ka dibna a xisaabaha yar oo sahlan na siiso formula ah si ay u xisaabiso wadarta horumarka ugu horeeyay ee ka tirsan, kuwaas oo:

S n = - (amayno * q - z 1) / (1 - q).

Bedelaadda, in caanaha miraheeda hadal z amayno 1 * q ^ (n-1) si ay u helaan caanaha caddad labaad ee horumarka: n S = - z1 * (q ^ n - 1) / (1 - q).

Waa istaahilaa dareenka xaqiiqada ah ka dib markii xiiso leh: kiniin dhoobo laga helay excavations ee Baabuloon qadiimiga ah, kaasi oo tilmaamaya VI ah. BC, waxaa ku jira hab cajiib ah wadarta 1 + 2 + ... + 22 + 29 loo siman yahay 2 ilaa laga jaray ay xoog u tobnaad 1. Sharaxa of this ifafaale aan weli la helay.

Waxaan la soco mid ka mid ah sifooyinka horumarka joomateri - shaqo joogta ah xubnaha, u dhaxeyso fogfog siman ka darafyadiisa oo isugu xigaan.

Muhiimad gaar ah marka xaga cilmiga of view, wax caynkaas ah sida horumarka joomateri koobi karayn iyo xisaabinta ay qadar. Haddii loo maleeyo in (yn) - a horumarka joomateri isagoo hooseeyaha q, dherginaya ah xaalad | q | <1, ay lacagta loo gudbin doonaa xadka dhinaca taas oo aan hore u ogaan wadarta xubnaha ugu horeysay, iyadoo kororka Fadli ah n, ka dibna waxay leeyihiin waxaa at dhowaanin xad la'aan.

Raadi qadarka this sabab u ah iyadoo la isticmaalayo qaaciddadan:

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

Iyo, sida waayo-aragnimo ayaa muujisay, waayo halleeyo lillaahinimada iyo Muuqda ee horumarka this Qarsoon iman kara codsi u weyn. Tusaale ahaan, haddii aan loo dhiso isku xigxiga oo ka mid ah wareegga ah sida ay isku geynta soo socda, isku Xidhka midpoints ka mid ah la soo dhaafay, ka dibna ay u sameeyaan horumar ah square koobi karayn joomateri isagoo hooseeyaha ah 1/2. foomka horumarka la mid ah iyo bedka saddex xagalada, heli marxalad kasta oo ka mid ah dhismaha, iyo wadarta waa loo siman yahay si ay meesha of square asalka ah.