Arithmetical phambili

Imisebenzi ye yokalo emileyo progression eyayikho kumaxesha amandulo. Ke wabonakala kusithiwa izisombululo, ngenxa yokuba yimfuneko practical.

Ngokomzekelo, kwenye lemibhalo waseYiputa yamandulo, ukuba isiqulatho zezibalo, - i-ngqangi Rhind (XIX BC) - iqulethe ngxaki: yahlula amanyathelo ezilishumi okuziinkozo abantu abalishumi, ngaphandle ukuba umahluko phakathi elowo kubo ngomnye-wesibhozo manyathelo ".

Kwaye kwiincwadi zezibalo yamaGrike amandulo, kukho theorems ezintle ezinxulumene an arithmetic inkqubela. Ngoko ke, Hypsicles Alexandria (II ngenkulungwane BC), imali eninzi imisebenzi umdla yaye wongezelela iincwadi elinesine ukuya 'ekuqaleni' of Euclid yaqulunqa ingcamango: "Xa yokalo emileyo zokuqhubela phambili nokuba inani lamalungu, isixa amalungu kwisiqingatha sesibini ngaphezu sum of amalungu 1 le yesibini ukuya iphindaphindeke isikweri 1/2 wamalungu. "

Sithatha inani engenasizathu of amanani zendalo (mkhulu zero), 1, 4, 7, ... n-1, n, ..., ekuthiwa ulandelelwano lwamanani.

Kubonisa ulandelelwano an. amanani ulandelelwano zibizwa ngokuba amalungu ayo yaye bakholisa zibonakaliswe iileta indices, abonisa inani lothotho yelungu (A1, A2, A3 ... funda: «okokuqala», «yesibini», «i-yokuhlamba 3" njalo-njalo ).

Ulandelelwano ingaba obungenasiphelo okanye sokuzalwa.

Kwaye yintoni izibalo phambili? It kuqondwa ngolandelelwano lwamanani afunyenwe ngokudibanisa ilungu langaphambili (n) kunye nenani elifanayo d, yona progression umahluko.

Ukuba d <0, ke thina sibe ukuqhubela ukuhla. Ukuba d> 0, ngoko eli inkqubela kucingwa ukuba ukwanda.

Arithmetic progression kuthiwa anesiphelo, xa sithatha kuphela nje ezimbalwa kwamalungu alo okuqala. Xa inani elikhulu kakhulu kumalungu iye kwisiqhoboshi zokuqhubela phambili.

Nayiphi progression arithmetic okunene uphiwa ngaye ifomula kulandelayo:

i = KN + b, ngoxa b kunye k - amanye amanani.

Ngokupheleleyo ntiyiso, nto leyo reverse: ukuba ulandelelwano unikwa ifomula efanayo, oko kanye zokuqhubela arithmetic, leyo iimpawu:

1. Ilungu ngalinye progression - imin lokubala eliguqukayo kweekota koko engaphambili.
2. : Ukuba, ukuqala owesibini, ilungu ngalinye - imin lokubala eliguqukayo kwikota engaphambili, kunye ezilandelayo, ie, ukuba le meko, oku ulandelelwano - an arithmetic zokuqhubela phambili. Le lokulingana zombini uphawu inkqubela, ke ngoko, lidla ngokubizwa ngokuba inkalo uphawu ukuqhubela phambili.
   Ngokufanayo, le theorem kuyinyaniso ukuba ibonisa lo propati: elandelwayo - an yokalo emileyo progression kuphela xa inxaki kuyinyaniso ukuba naziphi na amalungu ukulandelelana, kuqalwa owesibini.

A impahla uphawu naziphi amanani ngenxa ezine progression yokalo emileyo ukuze abonise am + = ak + al, ukuba m = k n + + l (m, n, k - inani progression).

Xa i arithmetic progression of naliphi (N-th) ilungu ofunekayo ingafunyanwa ngokusebenzisa ezi ndlela zilandelayo:

i = A1 + d (n-1).

Umzekelo: ilungu lokuqala (A1) ngendlela arithmetic progression unikwa kwaye ilingana ezintathu, kwaye umahluko (d) ilingana ezine. Fumana eziyimfuneko Ilungu lesihlanu-amane yale inkqubela. a45 = 1 + 4 (45-1) = 177

Formula an = ak + d (n - k) ukufumanisa n-th ixesha an yokalo emileyo kwinkqubela ngalinye ilungu layo k-th ezinikiweyo ukuba uyaziwa.

ngokwemiqathango Sum i yokalo emileyo progression (kokuthabatha amalungu n lokuqala anesiphelo progression) ibalwa ngale ndlela ilandelayo:

Sn = (A1 + i) n / 2.

Ukuba uyazi umahluko progression izibalo, kwaye ilungu yokuqala, ukuba ukubala nezinye ifomula luncedo:

Sn = ((2a1 + d (n-1)) / 2) * n.

Unyuselo-mali yokalo emileyo esiqulethe amalungu n, ibalwa ngolu hlobo lulandelayo:

Sn = (A1 + i) * n / 2.

ifomula Ukukhethwa yobalo ixhomekeke kwiimeko kunye iingxaki ze data ekuqaleni.

amanani Natural naliphi na inani ezifana 1,2,3, ..., n, ...- umzekelo elula an arithmetic zokuqhubela phambili.

Ukongeza kukho i izibalo progression kunye zejiyometri leyo anazo iimpawu kunye neempawu.