Geometric kev vam meej thiab nws cov khoom

Geometric kev vam meej yog ib qho tseem ceeb nyob rau hauv kev kawm txog zauv raws li ib tug science, thiab thov tseem ceeb, vim nws muaj ib tug tsis tshua sab tau, txawm nyob rau hauv lub kawm leb siab tshaj, piv txwv li, nyob rau hauv cov kev tshawb xav ntawm series. Tus thawj cov lus qhia txog cov kev kawm tau tuaj rau peb los ntawm ancient tim lyiv teb chaws, tshwj xeeb tshaj yog nyob rau hauv daim ntawv ntawm ib tug zoo-paub cov teeb meem ntawm lub Rhind papyrus xya neeg uas muaj xya lub miv. Variations ntawm no ua hauj lwm twb rov ntau lub sij hawm nyob rau txawv sij hawm los ntawm lwm haiv neeg. Txawm lub Velikiy Leonardo Pizansky, lub npe hu ua Fibonacci (XIII c.), Tau mus rau nws nyob rau hauv nws "Phau Ntawv tus Abacus."

Yog li ntawd hais tias cov duab kom vam meej muaj ib tug ancient keeb kwm. Nws nruab nrab yog ib tug hais txog zauv theem zuj zus nrog ib tug nonzero thawj neeg, thiab txhua tom ntej, pib nrog tus thib ob yog txiav txim los ntawm txawj rau yav tas los tsis tshua mob heev mis ntawm ib tug tsis tu ncua, nonzero xov tooj uas yog hu ua denominator kev vam meej (feem ntau yog nws xaiv siv tsab ntawv q).
Obviously, nws yuav pom los mus faib txhua tom ntej lub sij hawm ntawm ib theem zuj zus mus rau lub yav dhau los, i.e. z 2: z 1 = ... = ZN: z n-1 = .... Thiaj li, feem ntau txoj hauj lwm kom vam meej (ZN) txaus hais tias nws paub tus nqi ntawm cov thawj lub sij hawm ntawm lub denominator thiab y 1 q.

Piv txwv li, cia z 1 = 7, q = - 4 (q <0), ces cov hauv qab no geometric kev vam meej yog tau 7 - 28, 112 - 448, .... Raws li koj tau pom, lub resulting ib theem zuj zus yog tsis monotone.

Nco qab hais tias ib tug arbitrary ib theem zuj zus ntawm monotonous (ua / zuj zus lawm) thaum ib tug ntawm cov tswv cuab raws li ntau / tsawg tshaj li yav dhau los ib tug. Piv txwv li, ib theem zuj zus 2, 5, 9, ..., thiab -10, -100, -1000, ... - Monotone, tus thib ob tug - ib tug zuj zus lawm geometric kev vam meej.

Nyob rau hauv lub rooj plaub uas q = 1, tag nrho cov neeg muaj nyob tau, thiab nws yog hu ua lub qhov mob.

Cov kab yog qhov muaj mob ntawm no hom, nws yuav tsum los siav nram qab no tsim nyog thiab txaus mob, namely: pib los ntawm lub thib ob, txhua yam ntawm nws cov neeg yuav tsum tau cov duab phem ntawm neighboring neeg.

Qhov no tej khoom vaj tse tso cai nyob rau hauv tej yam ob uas nyob ib sab nrhiav arbitrary lub sij hawm muaj mob.

n-th sij hawm exponentially yooj yim nrhiav tau los ntawm cov mis: ZN = z 1 * q ^ (n-1), z paub thawj tus neeg 1 thiab denominator q.

Txij li thaum lub xov tooj ua ntu zus muaj ib tug zaum, ces ib tug ob peb yam yooj yim suav muab peb ib tug formula los mus laij lub sum ntawm cov thawj kev vam meej ntawm cov tswv cuab, namely:

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

Hloov, nyob rau hauv cov mis nws qhia tus nqi ZN z 1 * q ^ (n-1) kom tau ib tug thib ob sum mis ntawm lub kev vam meej: S n = - Z1 * (q ^ n - 1) / (1 - q).

Yog tsim nyog ntawm cov xim lub nram qab no nthuav qhov tseeb: lub av nplaum ntsiav tshuaj nyob rau hauv excavations ntawm ancient Babylon, uas yog hais txog mus rau lub VI. BC, muaj zoo kawg li txoj kev uas lub sum ntawm 1 + 2 + ... + 22 + 29 sib npaug zos rau 2 mus rau lub thib kaum hwj chim rho tawm 1. Cov kev tshab txhais ntawm no phenomenon tseem tsis tau pom.

Peb nco ntsoov ib tug ntawm cov yam ntxwv ntawm geometric kev vam meej - ib qhov chaw ua hauj lwm ntawm nws cov tswv cuab, spaced ntawm sib npaug zos deb ntawm qhov xaus ntawm ib theem zuj zus.

Ntawm kev tseem ceeb los ntawm ib tug scientific point of view, xws li ib tug tshaj plaws li ib txhis geometric kev vam meej thiab kev xam nws npaum li cas. Piv txwv tias yog hais tias (yn) - ib tug geometric kev vam meej muaj denominator q, satisfying tus mob | q | <1, nws ntau npaum li cas yuav tsum raug xa mus rau cov kev txwv ntawm uas peb twb paub cov sum ntawm nws xub, nrog unbounded nce ntawm n, ces muaj nws nce infinity.

Nrhiav tus nqi no raws li ib tug tshwm sim ntawm kev siv cov mis:

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

Thiab, raws li kev tau qhia, rau qhov khees simplicity ntawm no kom vam meej yog muab zais ib tug lossis loj daim ntawv thov tau. Piv txwv li, yog hais tias peb tsim ib tug sib lawv liag ntawm squares raws li cov nram qab no algorithm, txuas lub midpoints ntawm yav tas los ib tug, ces lawv tsim ib tug square infinite geometric kev vam meej muaj ib tug denominator 1/2. Cov tib kev vam meej daim ntawv thiab cheeb tsam ntawm cov vajvoos peb sab, tau nyob rau hauv txhua theem ntawm kev tsim kho, thiab nws sum yog sib npaug zos rau cov cheeb tsam ntawm tus thawj square.