Geometric zokuqhubela phambili kunye neempawu zayo

Geometric yenkqubela ebalulekileyo imathematika njengendlela inzululwazi, ndayinikela ukubaluleka, ekubeni ine- imihlaba ebanzi kakhulu, nkqu zemathematika eziphezulu, umzekelo, xa ngobhalo series. Ulwazi lokuqala kwinkqubela kwanawezayo kuthi, evela eYiputa yamandulo, ingakumbi ngokohlobo ingxaki ezaziwayo-kakuhle Rhind ngqangi abantu abasixhenxe kunye neekati ezisixhenxe. Imidlalo kwalo msebenzi phindwa amaxesha amaninzi ngamaxesha ahlukeneyo kwezinye iintlanga. Nkqu Leonardo Velikiy Pizansky, eyaziwa ngokuba Fibonacci (XIII c.), Wathetha naye wakhe "Book of abacus."

Ukuze ukuqhubela phambili zejometri une kwimbali yamandulo. Iluphawu ulandelelwano lwamanani kunye nelungu nonzero kuqala, kwaye elandelayo ngamnye, ukuqalela nge yesibini emiselwe phinda ifomula yokuphinda yangaphambili kwiziko, inani rhoqo nonzero into ekuthiwa progression idinomineyitha (it ngokuqhelekileyo okhethiweyo usebenzisa q unobumba).
Ngokucacileyo, ingafunyanwa ngokwahlula kwikota nganye elandelayo ngokulandelelana ukuya elidlulileyo, ngamanye z 2: z 1 = ... = Zn: z n-1 = .... Ngenxa yoko, kuba inkoliso progression umsebenzi (Zn) ezaneleyo ukuba uyayazi ixabiso kwekota yokuqala elingezantsi kunye y 1 q.

Ngokomzekelo, makhe z 1 = 7, q = - 4 (q <0), ngoko yenkqubela nezejometri ilandelayo ifunyenwe 7 - 28, 112 - 448, .... Njengoko ubona, ukulandelelana nto ayikho monotone.

Khumbula ukuba ulandelelwano engenasizathu of obungabangeli mdla (ukwanda / ukuhla) xa elinye lamalungu ayo alandela ngaphezulu / ngaphantsi kwe ezidlulileyo. Umzekelo, ukulandelelana 2, 5, 9, ..., ne -10, -100, -1000, ... - Monotone, Eyesibini - a liyehla zejometri ukuqhubela phambili.

Kwimeko apho q = 1, onke amalungu abafunyenwe ukuba, yaye kuthiwa rhoqo ukuqhubela phambili.

Ulandelelwano yaba ukuqhubela phambili olu hlobo, kufuneka anelise imeko efunekayo ezaneleyo zilandelayo, ezizezi: ukuqala wesibini, ngamnye amalungu ayo ukuba ibe imin zejometri yamalungu angabamelwane.

Lo mhlaba ivumela phantsi ezimbini ezingaqondakaliyo osecaleni progression zexeshana ngokwesigqibo esicalanye.

n-th elide ngokucacileyo lula yafunyanwa ifomula: Zn = z 1 * q ^ (n-1), z esazi ilungu yokuqala 1 kunye q eliphezulu nenani eliphantsi.

Ekubeni ukulandelelana inani uba mali, ngoko ezimbalwa izibalo ezilula kusinika ifomula ukubala inani progression yokuqala amalungu, ezizezi:

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

Ukutshintshwa, kwifomula layo ixabiso ibinzana Zn z 1 * q ^ (n-1) ukuze ufumane ifomula sum yesibini progression: S n = - Z1 * (q ^ n - 1) / (1 - q).

Ngaba ufanele ingqalelo into ilandelayo umdla: Icwecwe lodongwe i afumaneka bemba aseBhabhiloni yamandulo, elibhekisela VI. BC, iqulathe ngendlela emangalisayo udibaniso 1 + 2 + ... + 22 + 29 ilingana no-2 ukuya amandla thabatha yeshumi 1. Le ingcaciso yale nto ke iye okwangoku zifunyenwe.

Siyaphawula omnye iimpawu zejometri progression - umsebenzi rhoqo samalungu ayo, bebekwe kwi imigama olulinganayo evela eziphelweni ngokulandelelana.

Eyona nto ibaluleke ngokukodwa ukusuka kwindawo umbono lwenzululwazi, ikho into kwisiqhoboshi zejometri zokuqhubela phambili nokubala imali yayo. Sazigqibela ukuba (yn) - a zejometri progression ukuba idinomineyitha q, ukwanelisa imeko | q | <1, imali yayo iya kuthunyelwa kumda ekufuneka kakade inani lamalungu ayo yokuqala, kunye ukwanda olungathethekiyo n, ngoko unayo kuyo esondela obungapheliyo.

Fumana le mali ngenxa usebenzisa ifomula:

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

Ke, njengoko amava lubonise, kuba lula elikhangeleka le inkqubela ifihlakeleyo ibe isicelo enkulu. Umzekelo, ukuba akha ukulandelelana zezikwere ngokwezi algorithm ilandelayo, odibanisa iincam kowangaphambili, ngoko ke zakha sikwere ongapheliyo zejiyometri progression ukuba idinomineyitha 1/2. Ifom progression elifanayo kwindawo zoonxantathu, wafumana kwinqanaba ngalinye yokwakha, kwaye isiphumo sayo lilingana indawo yembutho yokuqala.