Mienzaniso yeMibvunzo Inokurukura nezveArithmetic Series
Kutevedzana kwemasvomhu ipfungwa huru mumasvomhu inowanzoonekwa mumatambudziko akasiyana-siyana, muchikoro chesekondari nedzidzo yepamusoro. Pfungwa iyi inosanganisira kutevedzana kwemanhamba apo temu yega yega imhedzisiro yekuwedzera kana kubvisa nhamba isingachinji kubva mutemu yapfuura. Muchinyorwa chino, tichakurukura mienzaniso yakati wandei yematambudziko nemhinduro dzawo kuti tinzwisise zvakanyanya pfungwa yekutevedzana kwemasvomhu.
Kunzwisisa Masvomhu Akatevedzana
Chitsauko chearithmetic inyaya ine musiyano unogara uripo pakati pemashoko maviri akatevedzana. Semuenzaniso, kana chitsauko chearithmetic chine izwi rekutanga \(a\) uye musiyano \(d\), saka mazwi acho anogona kunyorwa seizvi:
\[ a, a + d, a + 2d, a + 3d, \ldots \]
Kana tichida kuwana nth term yenhevedzano iyi, fomura yenth term (\(U_n\)) ndeiyi:
\[ U_n = a + (n-1)d \]
Zvichakadaro, huwandu hwemashoko ekutanga e n e arithmetic series (\(S_n\)) hunogona kuverengerwa uchishandisa fomura:
\[ S_n = \frac{n}{2} (2a + (n-1)d) \]
Mibvunzo yemuenzaniso nekukurukurirana
Muenzaniso Mubvunzo 1
Mubvunzo: Yakapihwa nhamba dzemasvomhu dzine izwi rekutanga \(a = 5\) uye musiyano wakajairika \(d = 3\). Tsvaga izwi rechigumi remutsara.
Kukurukurirana:
Kuti tiwane izwi rechigumi (\(U_{10}\)), tinogona kushandisa fomura yezwi rechigumi:
\[ U_{10} = a + (10-1)d \]
\[ U_{10} = 5 + (9 \cdot 3) \]
\[ U_{10} = 5 + 27 \]
\[ U_{10} = 32 \]
Saka, chikamu chegumi chechikamu ichi ndechemakumi matatu nemaviri.
Muenzaniso Mubvunzo 2
Mubvunzo: Tsvaga huwandu hwemashoko gumi nemashanu ekutanga eongororo yemasvomhu ane izwi rekutanga riri \(a = 4\) uye musiyano wakajairika uri \(d = 7\).
Kukurukurirana:
Kuti tiwane huwandu hwemashoko gumi nemashanu ekutanga (\(S_{15}\)), tinogona kushandisa fomura yehuwandu hwemashoko ekutanga n:
\[ S_{15} = \frac{15}{2} (2a + (15-1)d) \]
\[ S_{15} = \frac{15}{2} (2 \cdot 4 + 14 \cdot 7) \]
\[ S_{15} = \frac{15}{2} (8 + 98) \]
\[ S_{15} = \frac{15}{2} \cdot 106 \]
\[ S_{15} = 15 \cdot 53 \]
\[ S_{15} = 795 \]
Saka, huwandu hwemashoko gumi nemashanu ekutanga emutsara uyu i795.
Muenzaniso Mubvunzo 3
Mubvunzo: Zvinozivikanwa kuti chikamu chechishanu chechikamu chekuverenga masvomhu i20 uye chikamu chechi12 i48. Tsvaga chikamu chekutanga (\(a\)) uye musiyano wakajairika (\(d\)) wechikamu.
Kukurukurirana:
Kubva pane zvakapihwa mamiriro:
\[ U_5 = a + 4d = 20 \]
\[ U_{12} = a + 11d = 48 \]
Tine maequations maviri ane mitsara ine mavariable maviri atinogona kugadzirisa:
1. \( a + 4d = 20 \)
2. \( a + 11d = 48 \)
Kubva pa equation 1, tinogona kuratidza \(a\) maererano ne \(d\):
\[a = 20 – 4d \]
Zvino tava kutsiva \(a\) ne equation 2:
\[20 – 4d + 11d = 48 \]
\[ 20 + 7d = 48 \]
\[7d = 28 \]
\[d = 4 \]
Iye zvino tava kutsiva kukosha kwe \(d\) mu equation \(a = 20 - 4d\):
\[a = 20 – 4 \cdot 4 \]
\[a = 20 – 16 \]
\[ a = 4 \]
Saka, temu yekutanga yenhevedzano iyi i4 uye mutsauko wakajairika i4.
Muenzaniso Mubvunzo 4
Mubvunzo: Mangani mazwi anodiwa pakuverenga masvomhu ane temu yekutanga \(a = 2\) uye musiyano wakafanana \(d = 5\) kuti asvike pa200?
Kukurukurirana:
Muchiitiko ichi, tinofanira kuwana huwandu hwemashoko ekutanga e n (\(S_n\)) akaenzana ne200. Shandisa fomura yehuwandu hwemashoko ekutanga e n:
\[ S_n = \frac{n}{2} (2a + (n-1)d) = 200 \]
Tsiva kukosha kwe \(a\) uye \(d\):
\[ \frac{n}{2} (2 \cdot 2 + (n-1) \cdot 5) = 200 \]
\[ \frac{n}{2} (4 + 5n – 5) = 200 \]
\[ \frac{n}{2} (5n – 1) = 200 \]
\[ n (5n – 1) = 400 \]
Iyi iequation ine quadratic. Kuti tiigadzirise, tinochinja chimiro chayo:
\[ 5n^2 – n – 400 = 0 \]
Shandisa fomura yequadratic \(ax^2 + bx + c = 0\):
\[ n = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Muchiitiko ichi \(a = 5\), \(b = -1\), uye \(c = -400\):
\[ n = \frac{-(-1) \pm \sqrt{(-1)^2 – 4 \cdot 5 \cdot (-400)}}{2 \cdot 5} \]
\[ n = \frac{1 \pm \sqrt{1 + 8000}}{10} \]
\[ n = \frac{1 \pm \sqrt{8001}}{10} \]
Kukosha kwe \(\sqrt{8001}\) kuri pedyo ne89.42, saka:
\[ n = \frac{1 \pm 89.42}{10} \]
Tinotora tsika dzakanaka:
\[ n = \frac{1 + 89.42}{10} \]
\[ n \approx \frac{90.42}{10} \]
\[ n \inenge 9.042 \]
Saka, nhamba yemashoko anodiwa i9 (kana akatenderedzwa).
Mhedziso
Kutevedzana kwemasvomhu inyaya inokosha mumasvomhu. Kunzwisisa zvakakwana temu yekutanga, mutsauko wakajairika, temu yenth, uye huwandu hwemashoko ekutanga n zvakakosha pakugadzirisa matambudziko akasiyana-siyana. Nekushandisa mienzaniso nehurukuro dziri pamusoro, zvinotarisirwa kuti vaverengi vachanzwisisa zviri nani pfungwa huru dzekutevera kwemasvomhu.
