Cov Lus Nug Piv Txwv Sib Tham Txog Cov Kab Ke thiab Cov Series
Cov kab zauv thiab cov lej sib lawv liag yog cov ntsiab lus tseem ceeb hauv kev lej, uas feem ntau pom txij thaum kawm ntawv qib pib mus txog rau tsev kawm qib siab. Ib kab zauv yog cov lej sib sau ua ke uas tau teeb tsa raws li ib txoj cai tshwj xeeb, thaum ib kab zauv yog qhov sib sau ua ke ntawm cov nqe lus ntawm kab zauv ntawd. Hauv tsab xov xwm no, peb yuav tham txog ntau qhov teeb meem piv txwv thiab tham txog cov kab zauv thiab cov lej sib lawv liag.
Piv txwv 1: Kev suav lej sib lawv liag
Lo lus nug:
Muab ib qho lej sib law liag nrog thawj lo lus (a) = 3 thiab qhov sib txawv (d) = 5. Txheeb xyuas:
1. Lub sij hawm thib 10 ntawm cov kab zauv.
2. Qhov sib ntxiv ntawm 20 nqe lus thawj zaug ntawm cov kab zauv.
Kev Sib Tham:
1. Lub sijhawm kawm thib 10
Cov mis rau lub sijhawm nth ntawm cov lej suav yog:
\[
U_n = a + (n-1)d
\]
Rau lub sij hawm kawm thib 10 (U_10):
\[
U_{10} = 3 + (10-1) 5 = 3 + 45 = 48
\]
2. Tag Nrho ntawm 20 Lub Nqe Lus Thawj Zaug
Cov mis rau qhov sib ntxiv ntawm thawj n nqe lus (S_n) ntawm ib qho lej sib law liag yog:
\[
S_n = \frac{n}{2} (2a + (n-1)d)
\]
Rau qhov sib ntxiv ntawm 20 lo lus thawj zaug (S_20):
\[
S_{20} = \frac{20}{2} (2 \cdot 3 + (20-1) \cdot 5) = 10 (6 + 95) = 10 \cdot 101 = 1010
\]
Piv txwv 2: Cov Qauv Geometric
Lo lus nug:
Muab ib qho geometric sequence nrog thawj lo lus (a) = 4 thiab qhov ratio (r) = 2. Txheeb xyuas:
1. Lub sij hawm thib 6 ntawm cov kab zauv.
2. Qhov sib ntxiv ntawm 8 nqe lus thawj zaug ntawm cov kab zauv.
Kev Sib Tham:
1. Lub sijhawm kawm thib 6
Cov mis rau lub sijhawm nth ntawm cov kab ke geometric yog:
\[
U_n = a^{(n-1)}
\]
Rau lub sij hawm kawm thib 6 (U_6):
\[
U_{6} = 4 \cdot 2^{(6-1)} = 4 \cdot 2^5 = 4 \cdot 32 = 128
\]
2. Tag Nrho ntawm 8 Lub Nqe Lus Thawj Zaug
Cov mis rau qhov sib ntxiv ntawm thawj n nqe lus (S_n) ntawm cov kab ke geometric yog:
\[
S_n = ib \frac{r^n – 1}{r – 1}
\]
Rau qhov sib ntxiv ntawm 8 lo lus thawj zaug (S_8):
\[
S_{8} = 4 \frac{2^8 – 1}{2 – 1} = 4 \frac{256 – 1}{1} = 4 \cdot 255 = 1020
\]
Piv txwv 3: Convergent Geometric Infinite Series
Lo lus nug:
Muab ib qho geometric series nrog thawj lo lus (a) = 1 thiab qhov piv (r) = 1/2. Txheeb xyuas qhov sib ntxiv ntawm cov series infinite.
Kev Sib Tham:
Cov mis rau qhov sib ntxiv ntawm cov koob tsis kawg (S_∞) ntawm cov koob geometric sib sau ua ke yog:
\[
S_{\infty} = \frac{a}{1 – r}
\]
Yog li ntawd, rau qhov series no:
\[
S_{\infty} = \frac{1}{1 – \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2
\]
Piv txwv 4: Cov kab ke thiab cov lej ntawm cov lej plaub fab
Lo lus nug:
Muab ib kab zauv ntawm cov lej plaub fab nrog thawj kab lus (U_1) = 1, kab lus thib ob (U_2) = 4, thiab kab lus thib peb (U_3) = 9. Txheeb xyuas kab lus thib 5 ntawm kab lus. Kab lus no puas yog kab zauv lossis kab zauv geometric? Piav qhia.
Kev Sib Tham:
1. Lub sijhawm kawm thib 5
Tus qauv ntawm cov lej square yog:
\[
U_n = n^2
\]
Rau lub sij hawm kawm thib 5 (U_5):
\[
U_{5} = 5^2 = 25
\]
2. Hom Kab
Yuav xyuas seb qhov kev sib lawv liag no puas yog lej lossis geometric, xyuas qhov sib txawv ntawm cov nqe lus (qhov sib txawv sib xws) thiab qhov sib piv ntawm cov nqe lus:
– Qhov txawv ntawm cov lus (d):
\[
U_2 – U_1 = 4 – 1 = 3 \\
U_3 – U_2 = 9 – 4 = 5
\]
Vim tias qhov sib txawv tsis ruaj khov, qhov kev sib lawv liag no tsis yog lej.
– Qhov sib piv ntawm cov pab pawg neeg (r):
\[
\frac{U_2}{U_1} = \frac{4}{1} = 4 \\
\frac{U_3}{U_2} = \frac{9}{4} = 2.25
\]
Vim tias qhov sib piv ntawm cov nqe lus tsis ruaj khov, qhov kev sib lawv liag no tsis yog geometric.
Yog li, cov kab ke ntawm cov lej square no tsis yog lej lej lossis geometric sequence, tab sis yog ib qho sequence tshwj xeeb uas ua raws li tus qauv ntawm cov lej square.
Piv txwv 5: Cov lej suav tsis kawg
Lo lus nug:
Puas muaj peev xwm suav qhov sib ntxiv ntawm cov lej suav tsis kawg? Yog tias muaj, muab ib qho piv txwv. Yog tias tsis muaj, piav qhia vim li cas.
Kev Sib Tham:
Tsis zoo li cov lej geometric, cov lej suav tsis kawg feem ntau tsis muaj qhov sib npaug kawg. Qhov no yog vim tias txhua lo lus nce lossis txo qis linearly, yog li qhov sib npaug txuas ntxiv mus tas li.
Piv txwv li, xav txog ib qho kev suav lej tsis kawg nrog thawj lo lus 1 thiab qhov sib txawv 1:
\[
1 + 2 + 3 + 4 + \ldots
\]
Yog tias peb sim suav lawv ua ke, nws pom tseeb tias cov koob yuav tsis sib sau ua ke rau tus nqi ruaj khov, tab sis yuav mus txog qhov tsis kawg. Yog li ntawd, qhov sib sau ua ke ntawm cov koob lej tsis kawg yog, feem ntau, tsis kawg thiab tsis tuaj yeem suav tau zoo li cov koob lej sib sau ua ke.
-
Hauv tsab xov xwm no, peb tau tham txog ntau yam teeb meem piv txwv thiab tham txog cov kab ke thiab cov koob. Peb tau tshuaj xyuas cov lej lej thiab cov kab ke geometric, pom yuav ua li cas xam cov lus nth thiab qhov sib ntxiv ntawm lawv cov lus thawj zaug, thiab teb cov lus nug txog cov koob tsis kawg. Los ntawm kev nkag siab cov ntsiab lus yooj yim thiab cov piv txwv no, peb vam tias koj yuav muaj kev ntseeg siab ntau dua thaum mus txog cov kab ke thiab koob hauv lej.
