Qhov Sib Txawv Ntawm Cov lej lej thiab Geometric Sequence

2024 Tus sau: Alex Aldridge | [email protected]. Kawg hloov kho: 2024-01-17 23:34

lej lej vs Geometric Sequence

Kev kawm txog cov qauv ntawm cov lej thiab lawv tus cwj pwm yog ib qho kev kawm tseem ceeb hauv kev ua lej. Feem ntau cov qauv no tuaj yeem pom hauv qhov xwm txheej thiab pab peb piav qhia lawv tus cwj pwm hauv qhov kev xav ntawm kev tshawb fawb. Arithmetic sequences thiab Geometric sequences yog ob qho ntawm cov qauv yooj yim uas tshwm sim hauv cov lej, thiab feem ntau pom nyob rau hauv tej yam tshwm sim.

Cov kab ke yog cov lej xaj. Tus naj npawb ntawm cov khoom nyob rau hauv ib ntus tuaj yeem yog qhov kawg lossis tsis kawg.

Ntau txog lej lej (Arithmetic Progression)

Ib qho lej lej txhais tau tias yog ib ntu ntawm cov lej nrog qhov sib txawv tas li ntawm txhua lub sijhawm sib law liag. Nws tseem yog lub npe hu ua kev ua lej lej.

Arithmetic Sequnece ⇒ a₁, a₂, a_3, a ₄ , …, a_n; where a₂ =a₁ + d, a₃ =a_{2+ d, thiab lwm yam.}

Yog thawj lo lus yog a₁ thiab qhov sib txawv yog d, ces tus n^{th lub sij hawm ntawm qhov sib lawv liag yog muab los ntawm;}

a_n =a_{1 + (n-1)d}

Los ntawm kev noj cov txiaj ntsig saum toj no ntxiv, n^{th lub sijhawm tuaj yeem muab tau raws li;}

a_n =a_m + (n-m)d, qhov twg a_{m yog ib lo lus random > m.}

Cov txheej ntawm cov lej thiab cov lej khib yog cov piv txwv yooj yim ntawm cov lej lej, qhov twg txhua ntu muaj qhov sib txawv (d) ntawm 2.

Tus lej ntawm cov ntsiab lus hauv ib ntu tuaj yeem ua tsis kawg lossis tsis kawg. Nyob rau hauv cov ntaub ntawv infinite (n → ∞), qhov sib lawv liag nyhav rau infinity nyob ntawm qhov sib txawv (a_{n→ ± ∞). Yog tias qhov sib txawv sib txawv yog qhov zoo (d > 0), qhov sib lawv liag nyhav rau qhov zoo infinity thiab, yog tias qhov sib txawv tsis zoo (d < 0), nws nyhav rau qhov tsis zoo infinity. Yog hais tias cov nqe lus yog finite, qhov sib lawv liag kuj yog finite.}

Cov ntsiab lus hauv cov lej lej yog lub npe hu ua lej lej: S_n=a₁ + a ₂ + a₃ + a₄ + ⋯ + a_n =∑ i=1→n_ai; _{and S}n_{=(n/2) (a}1 _{+ a}n_{)=(n/2) [2a}1 _{+ (n-1)d] muab tus nqi ntawm qhov series (S}n)

Ntau txog Geometric Sequence (Geometric Progression)

A geometric sequence yog txhais raws li ib ntus uas qhov quotient ntawm ob nqe lus sib law liag yog qhov tas mus li. Qhov no tseem hu ua geometric progression.

Geometric sequence ⇒ a₁, a₂, a₃, a₄ , …, a_n; where a₂/a₁=r, a₃/a_{2=r, thiab lwm yam, qhov twg r yog tus lej tiag.}

Nws yooj yim dua los sawv cev rau cov kab ke geometric siv qhov sib piv (r) thiab lub ntsiab lus pib (a). Li no geometric sequence ⇒ a₁, a₁r, a₁r^{2, a}1_r3^{, …, a}1_r.

Ntawm daim ntawv ntawm n^th cov ntsiab lus muab los ntawm a_n =a₁r ^n-1. (Poob subscript of the first term ⇒ a_n =ar^n-1)

Cov kab ke geometric kuj tuaj yeem ua tiav lossis tsis kawg. Yog hais tias tus naj npawb ntawm cov nqe lus yog finite, qhov sib lawv liag yog hais tias yuav finite. Thiab yog hais tias cov nqe lus yog infinite, qhov sib lawv liag yuav yog infinite los yog finite nyob ntawm tus piv r. Qhov sib piv feem ntau cuam tshuam rau ntau yam khoom hauv geometric sequences.

r > o	0 < r < +1	Cov kab ke sib hloov - kev nthuav tawm kev puas tsuaj, piv txwv li an → 0, n → ∞
	r=1	Constant sequence, i.e. an=constant
	r > 1	Qhov Kev Sib Tw sib txawv - kev loj hlob exponential, i.e. an→ ∞, n → ∞
r < 0	-1 < r < 0	Cov kab ke yog oscillating, tab sis converges
	r=1	Cov kab ke sib hloov thiab tsis tu ncua, i.e. an=±constant
	r < -1	Cov kab ke sib hloov thiab sib txawv. i.e. an → ±∞, n → ∞
r=0	qhov sib lawv liag yog ib txoj hlua ntawm xoom

N. B: Nyob rau hauv txhua rooj plaub saum toj no, a₁ > 0; yog tias a₁ < 0, cov cim ntsig txog a_{n yuav hloov pauv.}

Lub sijhawm ncua sij hawm ntawm lub pob bounces ua raws li geometric ib ntus hauv cov qauv zoo tagnrho, thiab nws yog qhov sib xyaw ua ke.

Cov lej ntawm cov ntsiab lus ntawm geometric sequence yog hu ua geometric series; S_n =ar+ ar² + ar³ + ⋯ + arⁿ=∑_{i=1 →n} ar^{i. Cov lej ntawm cov lej geometric tuaj yeem suav nrog cov qauv hauv qab no.}

S_n =a(1-r^{)/(1-r); qhov twg a yog thawj lo lus thiab r yog qhov piv.}

Yog qhov piv, r ≤ 1, cov koob sib txuas. Rau ib qho tsis kawg, tus nqi ntawm kev sib koom ua ke yog muab los ntawm S_n=a/(1-r)

Dab tsi yog qhov txawv ntawm Arithmetic thiab Geometric Sequence/Progression?

• Hauv kev lej lej, ob nqe lus sib law liag muaj qhov sib txawv (d) thaum, nyob rau hauv geometric sequence, ob nqe lus sib law liag muaj qhov sib npaug (r).

• Hauv kev ua lej lej, qhov sib txawv ntawm cov ntsiab lus yog linear, piv txwv li ib txoj kab ncaj tuaj yeem kos dhau los ntawm tag nrho cov ntsiab lus. Nyob rau hauv ib tug geometric series, qhov variation yog exponential; los yog loj hlob los yog lwj raws li qhov sib piv.

• Tag nrho cov lej lej tsis kawg yog qhov sib txawv, whereas infinite geometric series tuaj yeem sib txawv lossis sib xyaw.

• Cov kab geometric tuaj yeem qhia oscillation yog qhov piv r yog qhov tsis zoo thaum cov lej lej tsis pom oscillation