Cov Lus Nug Piv Txwv Sib Tham Txog Cov Txheej Txheem Xam Xaj
Cov kab zauv yog ib lub tswv yim tseem ceeb hauv kev lej uas feem ntau tshwm sim hauv ntau yam teeb meem, ob qho tib si hauv tsev kawm ntawv theem nrab thiab kev kawm qib siab. Lub tswv yim no suav nrog cov kab zauv uas txhua lo lus yog qhov tshwm sim ntawm kev ntxiv lossis rho tawm tus lej tas mus li los ntawm lo lus dhau los. Hauv tsab xov xwm no, peb yuav tham txog ntau qhov teeb meem piv txwv thiab lawv cov kev daws teeb meem kom nkag siab ntxiv txog lub tswv yim ntawm cov kab zauv.
Kev Nkag Siab Txog Cov Qauv Xam Xaj
Ib qho kev suav lej yog ib qho kev suav lej uas muaj qhov sib txawv tas li (qhov sib txawv) ntawm ob lo lus sib law liag. Piv txwv li, yog tias ib qho kev suav lej muaj thawj lo lus \(a\) thiab qhov sib txawv \(d\), ces cov lus tuaj yeem sau raws li hauv qab no:
\[ ib, ib + d, ib + 2d, ib + 3d, \ldots \]
Yog tias peb xav nrhiav lub nth term ntawm cov series no, cov mis rau lub nth term (\(U_n\)) yog:
\[ U_n = a + (n-1)d \]
Lub caij no, qhov sib ntxiv ntawm thawj n nqe lus ntawm cov lej suav (\(S_n\)) tuaj yeem suav los ntawm kev siv cov mis:
\[ S_n = \frac{n}{2} (2a + (n-1)d) \]
Cov Lus Nug Piv Txwv thiab Kev Sib Tham
Piv txwv lus nug 1
Lus Nug: Muab ib qho lej suav nrog thawj lo lus \(a = 5\) thiab qhov sib txawv sib xws \(d = 3\). Nrhiav lo lus thib 10 ntawm cov series.
Kev Sib Tham:
Yuav nrhiav tau lo lus thib 10 (\(U_{10}\)), peb siv tau cov mis nth lo lus:
\[ U_{10} = a + (10-1)d \]
\[ U_{10} = 5 + (9 \cdot 3) \]
\[ U_{10} = 5 + 27 \]
\[ U_{10} = 32 \]
Yog li, lub sijhawm thib 10 ntawm cov koob yog 32.
Piv txwv lus nug 2
Lus Nug: Nrhiav qhov sib ntxiv ntawm 15 lo lus thawj zaug ntawm cov lej suav uas nws lo lus thawj zaug yog \(a = 4\) thiab qhov sib txawv sib xws yog \(d = 7\).
Kev Sib Tham:
Yuav nrhiav tau qhov sib ntxiv ntawm 15 lo lus thawj zaug (\(S_{15}\)), peb siv tau cov mis rau qhov sib ntxiv ntawm thawj n lo lus:
\[ S_{15} = \frac{15}{2} (2a + (15-1)d) \]
\[ S_{15} = \frac{15}{2} (2 \cdot 4 + 14 \cdot 7) \]
\[ S_{15} = \frac{15}{2} (8 + 98) \]
\[ S_{15} = \frac{15}{2} \cdot 106 \]
\[ S_{15} = 15 \cdot 53 \]
\[ S_{15} = 795 \]
Yog li, qhov sib ntxiv ntawm 15 nqe lus thawj zaug ntawm cov koob yog 795.
Piv txwv lus nug 3
Lus Nug: Paub tias tus lej thib 5 ntawm cov lej suav yog 20 thiab tus lej thib 12 yog 48. Nrhiav tus lej thawj zaug (\(a\)) thiab qhov sib txawv (\(d\)) ntawm cov lej no.
Kev Sib Tham:
Los ntawm cov xwm txheej tau muab:
\[ U_5 = a + 4d = 20 \]
\[ U_{12} = a + 11d = 48 \]
Peb muaj ob qho kev sib npaug linear nrog ob qho kev hloov pauv uas peb tuaj yeem daws tau:
1. \( a + 4d = 20 \)
2. \( a + 11d = 48 \)
Los ntawm kab zauv 1, peb tuaj yeem qhia \(a\) hauv cov nqe lus ntawm \(d\):
\[ ib = 20 – 4d \]
Tam sim no peb hloov \(a\) rau hauv kab zauv 2:
\[ 20 – 4d + 11d = 48 \]
\[ 20 + 7d = 48 \]
\[ 7d = 28 \]
\[ d = 4 \]
Tam sim no peb hloov tus nqi ntawm \(d\) rau hauv kab zauv \(a = 20 - 4d\):
\[ a = 20 – 4 \cdot 4 \]
\[ ib = 20 – 16 \]
\[ ib = 4 \]
Yog li, thawj lo lus ntawm cov koob yog 4 thiab qhov sib txawv yog 4.
Piv txwv lus nug 4
Lus Nug: Xav tau pes tsawg lo lus rau cov lej suav nrog thawj lo lus \(a = 2\) thiab qhov sib txawv \(d = 5\) kom suav tau 200?
Kev Sib Tham:
Hauv qhov no, peb yuav tsum nrhiav qhov sib ntxiv ntawm thawj n nqe lus (\(S_n\)) uas sib npaug rau 200. Siv cov mis rau qhov sib ntxiv ntawm thawj n nqe lus:
\[ S_n = \frac{n}{2} (2a + (n-1)d) = 200 \]
Hloov cov nqi ntawm \(a\) thiab \(d\):
\[ \frac{n}{2} (2 \cdot 2 + (n-1) \cdot 5) = 200 \]
\[ \frac{n}{2} (4 + 5n – 5) = 200 \]
\[ \frac{n}{2} (5n – 1) = 200 \]
\[ n (5n – 1) = 400 \]
Qhov no yog ib qho quadratic equation. Yuav kom daws tau nws, peb hloov nws daim ntawv:
\[ 5n^2 – n – 400 = 0 \]
Siv cov qauv quadratic \(ax^2 + bx + c = 0\):
\[ n = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Hauv qhov xwm txheej no \(a = 5\), \(b = -1\), thiab \(c = -400\):
\[ n = \frac{-(-1) \pm \sqrt{(-1)^2 – 4 \cdot 5 \cdot (-400)}}{2 \cdot 5} \]
\[ n = \frac{1 \pm \sqrt{1 + 8000}}{10} \]
\[ n = \frac{1 \pm \sqrt{8001}}{10} \]
Tus nqi ntawm \(\sqrt{8001}\) ze rau 89.42, ces:
\[ n = \frac{1 \pm 89.42}{10} \]
Peb siv cov txiaj ntsig zoo:
\[ n = \frac{1 + 89.42}{10} \]
\[ n \approx \frac{90.42}{10} \]
\[ n \kwv yees li 9.042 \]
Yog li, tus naj npawb ntawm cov lus xav tau yog 9 lo lus (yog tias puag ncig).
Xaus
Cov kab zauv yog ib qho tseem ceeb hauv kev kawm lej. Kev nkag siab zoo txog thawj lo lus, qhov sib txawv sib xws, nth term, thiab qhov sib ntxiv ntawm thawj n term yog qhov tseem ceeb heev rau kev daws ntau yam teeb meem. Los ntawm kev siv cov piv txwv thiab kev sib tham saum toj no, vam tias cov nyeem ntawv yuav nkag siab zoo dua txog cov ntsiab lus yooj yim ntawm cov kab zauv.
