Lissafi da kuma samun ci gaban. MISALI ga yanke shawara

La'akari da jere.

7 28 112 448 1792 ...

Quite a fili ya nuna cewa, darajar wani na abubuwa fiye da baya daidai da sau hudu. Saboda haka, wannan jerin ne a ci gaban.

geometrical ci gaban kira iyaka jerin lambobi, da babban alama wanda shi ne cewa wadannan da lambar da aka samu daga sama ta halitta da wasu tabbataccen adadin. Wannan yana bayyana da wadannan dabara.

_{mai z +1 = wani z · q} , inda z - lambar da aka zaba kashi.

Haka kuma, z ∈ N.

A lokacin da makaranta da aka yi karatu na lissafi ci gaban - 9th sa. Misalai zai taimaka fahimci manufar:

0.25 0.125 0.0625 ...

18 Fabrairu 6 ...

Bisa ga wannan dabara, da ci gaban na hada iya samu kamar haka:

Babu q, ko b _z ba zai iya zama sifili. Har ila yau, kowane daga cikin abubuwa na jerin lambobin ci gaban kada ta kasance sifili.

Haka kuma, ganin na gaba yawan lamba, ninka karshen ta q.

Don ayyana wannan ci gaban, dole ne ka saka na farko kashi na shi da kuma hada. Bayan da cewa yana yiwuwa a sami wani daga cikin wadannan mambobi da su adadin.

jinsunan

Dangane da q kuma a _1, wannan ci gaban ne zuwa kashi da dama iri:

• Idan wani _1, da kuma q ne mafi girma daga daya, sa'an nan a jerin - kara da kowane m kashi na wani lissafi da ci gaban. Misalai daga gare an daki-daki a kasa.

Misali: a ₁ = 3, q = 2 - mafi girma daga hadin kai, duka biyu sigogi.

Sa'an nan a jerin lambobi za a iya rubuta kamar:

3 6 12 24 48 ...

• Idan | q | kasa da daya, watau, shi ne daidai da multiplication da division, da ci gaban da irin wannan yanayi - ragewa lissafi ci gaban. Misalai daga gare an daki-daki a kasa.

Misali: a ₁ = 6, q = 1/3 - wani ₁ ne mafi girma daga daya, q - kasa.

Sa'an nan a jerin lambobi za a iya rubuta kamar haka:

6 2 2/3 ... - wani kashi mafi abubuwa wadannan da shi, shi ne sau 3.

• Alternating. Idan q <0, a cikin ãyõyin da lambobi daga cikin jerin alternating kullum ko da wani _1, da kuma abubuwa na wani karuwa ko raguwa.

Misali: a ₁ = -3, q = -2 - ne duka biyu kasa da sifiri.

Sa'an nan a jerin lambobi za a iya rubuta kamar:

3, 6, -12, 24, ...

dabara

Domin m amfani, akwai mutane da yawa na lissafi progressions na dabarbari:

• Formula z-th lokaci. Yana da damar lissafi na kashi a wani takamaiman adadin ba tare da kirga baya lambobi.

Misali: q = 3, a = ₁ 4. ake bukata don ƙidaya a karo na hudu kashi ci gaban.

Magani: wani = ₄ 4 3 · · ^4-1 3 = 4 ³ = 4 · 27 = 108.

• Jimlar na farko abubuwa, wanda lambar da yake daidai to z. Yana da damar lissafi na Naira Miliyan Xari da dukkan abubuwa a wani jerin zuwa wani _z m.

≠ 0, ta haka, q ba 1 - (q 1) Tun da (1- q) ne a hada, sa'an nan.

Lura: idan q = 1, sa'an nan da ci gaban zai yi wakilci a yawan karshe maimaita lamba.

Adadin dabam dabam misalai: a ₁ = 2, q = -2. Lissafi S _5.

Magani: S ₅ = 22 - lissafi dabara.

• Adadin idan | q | <1 da kuma lokacin da z o ƙarin tabbatar da rashin iyaka.

Misali: a ₁ = _2, q = 0.5. Nemo jimla.

Magani: S _z = 2 x = 4

Idan muka lissafta da Naira Miliyan Xari da dama mambobin manual, za ka ga cewa shi ne, haƙĩƙa, aikata zuwa hudu.

S _z = 1 + 2 + 0.5 + 0.25 + 0,125 + 0.0625 = 3.9375 4

Wasu Properties:

• A halayyar dukiya. Idan da wadannan yanayin Yana riqe ga wani z, sa'an nan ba na lamba jerin - wani lissafi da ci gaban:

mai _z ² = A _{z -1} · A _{z + 1}

• Shi ne kuma square daga wani adadin ne dabam dabam ta wajen Bugu da kari daga cikin murabba'ai da sauran biyu lambobi a cikin kowace jere, idan sun kasance equidistant daga kashi.

² wani _z = wani _{z - t} ² ₊ wani _z + _t ² inda t - nisa tsakanin wadannan lambobi.

• The abubuwa bambanta da q sau.
• A logarithms daga cikin abubuwa na ci gaban da ya samar da wata ci gaban, amma ilmin lissafi, da cewa shi ne, kowane daga cikinsu fiye da baya daya ta wani adadi.

Misalai na wasu gargajiya matsaloli

Don fi gane abin da wani lissafi da ci gaban, tare da yanke shawara misalai ga daraja 9 iya taimaka.

• Sharuɗɗa da: a ₁ = 3, a ₃ = 48. Nema q.

Magani: kowane m kashi a fiye da baya q lokaci. Wajibi ne a bayyana wasu abubuwa ta hanyar wasu via hada.

Saboda haka, a ₃ = q ² · ₁

Lokacin musanya q = 4

• Yanayi: ₂ = 6, a = ₃ 12. lissafi S _6.

Magani: Don yin wannan, shi isa a samu q, na farko kashi da kuma canza zuwa cikin dabara.

mai ₃ = q · _2, saboda haka, q = 2

₂ = q · A _1, saboda haka mai = ₁ 3

S = ₆ 189

• · A ₁ = 10, q = -2. Nemo na huɗu kashi na ci gaban.

Magani: shi ne isa zuwa ga bayyana na huɗu kashi ta hanyar da farko, kuma ta hanyar da maimaita lamba.

₄ a ³ = q · mai = ₁ -80

Aikace-aikace misali:

• Bankin abokin ciniki ya bayar da gudunmawar Naira Miliyan Xari da 10,000 rubles, a karkashin abin da a kowace shekara abokin ciniki zuwa babba adadin za a kara 6% na shi da yake. Nawa kudi ne a cikin asusun bayan 4 shekaru?

Magani: A farko adadin daidaita 10 dubu rubles. Saboda haka, shekara guda bayan da zuba jari a cikin asusun zai zama da adadin daidai to 10000 + 10000 = 10000 · · 0.06 1,06

Haka kuma, adadin a cikin account ko da bayan shekara guda zai iya bayyana kamar haka:

(10000 · 1,06) · 10000 · 0.06 + 1,06 = 1,06 · 1,06 · 10000

Wannan shi ne, kowace shekara yawan karu zuwa 1,06 sau. Saboda haka, samun yawan asusun bayan 4 shekaru, shi isa a samu a karo na hudu kashi ci gaban, wanda aka bai farko kashi daidaita 10 dubu, da kuma hada daidaita 1,06.

S = 1,06 · · 1,06 1,06 · 1,06 · 10000 = 12625

Misalan matsaloli a ƙidãyar Naira Miliyan Xari:

A daban-daban matsaloli amfani na lissafi ci gaban. Wani misali na gano da Miliyan Xari iya saita kamar haka:

wani ₁ = 4, q = 2, lissafi S _5.

Magani: duk dole data ga lissafi an san, kawai musanya su a cikin dabara.

S ₅ = 124

• ₂ = 6, a = ₃ 18. lissafi da Naira Miliyan Xari da farko shida abubuwa.

bayani:

A Geom. da ci gaba na kowane kashi na gaba ya fi girma fiye da na baya q sau, wato, yin lissafi da adadin da ka bukatar ka san kashi a ₁ da kuma hada q.

₂ · q = wani ₃

q = 3

Hakazalika, da bukatar su sami wani _{1, 2} da kuma sani q.

wani ₁ · q = ₂

wani ₁ = 2

Kuma a sa'an nan Ya isa ga musanya da aka sani data cikin dabara adadin.

S ₆ = 728.