Geometric Holomua a me kona waiwai

Geometric Holomua mea nui i loko o makemakika like me ka 'epekema, a noi aku la manaʻo nui, no ka mea, i ka loa ka laula laulā, a hiki ma ka makemakika kiekie ae, no ka mea hoike, i loko o ka pähola' o ka moʻo. Ka mua 'ike ma luna o ka holomua, hele mai ia mākou, mai kahiko Aigupita, pakahi aku la ia i loko o ka palapala o ka luawai-ike pilikia o ka Rhind papyrus ehiku kanaka me ehiku nā pōpoki. Likeʻole o kēia hana i hai na manawa he nui i kekahi mau manawa, mai nā lāhui kanaka. ʻO ka Velikiy Leonardo Pizansky,ʻikeʻia me Fibonacci (XIII c.), Olelo mai ia ia ia i loko o kona "Buke o ka Abacus."

No laila, i ka geometric Holomua mai he kahiko mōʻaukala. Ua ho i ka laulā kaʻina me ka nonzero mua lala, a me kēlā mahope, e hoʻomaka ana me ka lua me ka nānā 'ma ka hoonui i ka mua recurrence haʻilula ma ka ikaika, nonzero helu i ua kapaia denominator Holomua (ka mea ana i koho hoʻohana' ana i ka palapala Q).
ʻae nō hoʻi, ka mea hiki ke loaʻa ma ka māheleʻana i kēlā me kēia mahope manawa o ke kaʻina i ka mea mua, a laila Z 2: Z 1 = ... = zn: Z N-1 = .... Nolaila, no ka hapanui hana Holomua (zn) lawa i ka mea i ike i ka waiwai o ka mea mua kau o ka denominator a me y me 1 Q.

No ka laʻana, e Z 1 = 7, Q = - 4 (Q <0), a laila, i ka i kēia geometric Holomua ua loaa 7 - 28, 112 - 448, .... E like me 'oe ke ike, i ke kūpono kaʻina mea ole monotone.

Keia iloko o i ka ākeʻakeʻa kumu kaʻina o monotonous (hoʻoulu / ka hōʻemiʻana) wā i kekahi o kona mau lala ma hope oi / emi mai ma mua o ka mea mua kekahi. No ka laʻana, ke kaʻina 2, 5, 9, ..., a me -10, -100, -1000, ... - Monotone, i ka lua o kekahi - i ka hōʻemiʻana geometric Holomua.

Ma ka hihia kahi Q = 1, ua loaʻa i nā hoa a pau e ia, a me ka mea, ua kapaʻia ka ikaika Holomua.

Ke kaʻina i ka Holomua o kēia 'ano, ka mea e panai ai i na kēia mau mea e pono ai a me ka lawa ana, oia hoi: e hoʻomaka ana mai o ka lua, kela mea keia mea o kona mau lala e ia i ka geometric mawaena o kokoke lālā.

Kēia waiwai apono 'ma lalo o kekahi mau pili mea i hoʻoholo' ākeʻakeʻa kumu makahiki Holomua.

N-la manawa exponentially wale i loaʻa ma ka haʻilula: zn = Z 1 * Q ^ (N-1), Z ka ike mua lala 1 a me ka denominator Q.

Mai ka helu kaʻina mai i ka puu dala, a laila, he mau mea ho haawi mai ia makou i ka haʻilula e hoʻomaulia i ka huina o ka Holomua mua o nā lālā, oia hoi:

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

Waiho hou, ma ka haʻilula kona 'aoʻao o cia zn Z 1 * Q ^ (N-1) e loaa i ka lua o ka huina haʻilula o ka Holomua: S N = - z1 * (Q ^ N - 1) / (1 - Q).

Mea pono o ka hoʻolohe i nā kēia mau hana hoihoi mea: ka lepo papa i loaʻa i loko o hanaʻeliʻana o kahiko Babulona, a pili ia i ka VI. BC, he kupaianaha aoao o ka huina o ka 1 + 2 + ... + 22 + 29 like me 2 a hiki i ka umi o ka mana hua 1. Ka wehewehe ana o keia kŰia iʻaʻole naʻe, ua loaʻa.

E nana i kekahi o ka waiwai o geometric Holomua - he hana ikaika o kona mau lala, kaulike ma like ka mamao, mai na kihi aku o ke kaʻina.

A mau mea nui, mai ka 'epekema wahi o ka hoomaopopo, e like me ka mea me he mana loa geometric Holomua a me ka helu ana kona uku. Holoholo ana (yn) - he geometric Holomua ka denominator Q, satisfying ke ano | Q | <1, kona dala e e haawiia i ka palena i a mākou i mua o ka huina o kona mau lala mua, me ka unbounded māhuahua o ka n, alaila, i ma ia kokoke infinity.

E huli i keia dala me ka hopena o ka hoʻohana 'ana i ka haʻilula:

S N = y me 1 / (1- Q).

A, me ka hoao ana i hoike mai, no ka mea ka nohoaliʻi ke akahai o keia Holomua ua hūnāʻia i ka nui palapala noi Loaʻaʻia. No ka laʻana, ina ua kükulu i kekahi kaʻina o nā kihikihiʻo e like me ka hahai ana algorithm, ai pili aku i ka midpoints o ka mea mua kekahi, a laila ka mea i hana i ka huinahalike mana loa geometric Holomua me ke denominator 1/2. Ka Ia Holomua kino a me ka wahi o ka triangle mauna, loaa ma kēlā me kēia kahua o ke kūkuluʻia, a me kona huina mea like me ka wahi o ka palapala huinahalike.