Mibvunzo Yemuenzaniso Yekukurukura Kutevedzana KwemaArithmetic
Kutevedzana kwemasvomhu ipfungwa huru mumasvomhu inowanzoonekwa mubvunzo dzakasiyana-siyana uye mashandisirwo chaiwo. Kutevedzana kwemasvomhu kutevedzana kwemanhamba kune musiyano unogara uripo pakati pemashoko maviri akatevedzana. Muchinyorwa chino, tichaongorora pfungwa yekutevedzana kwemasvomhu zvakadzama kuburikidza nematambudziko akati wandei ane tsananguro dzakakwana.
Tsanangudzo neMagwaro
Tisati tapinda mumatambudziko emuenzaniso, zvakakosha kunzwisisa notation inowanzoshandiswa mu ma arithmetic sequences. Kana \(a\) iri izwi rekutanga uye \(d\) iri common difference (constant difference), saka ma arithmetic sequence inogona kunyorwa seizvi:
\[ a, a+d, a+2d, a+3d, \ldots, a+(n-1)d \]
Izwi rechina (Un) remutsara uyu rinogona kunyorwa seizvi:
\[ U_n = a + (n-1)d \]
Inotevera mienzaniso yakati wandei yemibvunzo nehurukuro dzayo kuti zvive nyore kunzwisisa masvomhu.
Muenzaniso Mubvunzo 1
Mubvunzo:
Zvichipiwa nhamba yekuverenga ine izwi rekutanga \(a = 5\) uye musiyano wakajairika \(d = 3\). Sarudza izwi rechigumi renhamba.
Kukurukurirana:
Kushandisa fomura yakajairika yetemu nth, iyo \( U_n = a + (n-1)d \):
\[ U_{10} = 5 + (10-1) \cdot 3 \]
\[ U_{10} = 5 + 9 \cdot 3 \]
\[ U_{10} = 5 + 27 \]
\[ U_{10} = 32 \]
Saka, chikamu chechishanu chemutsara uyu i32.
Muenzaniso Mubvunzo 2
Mubvunzo:
Zvichipiwa kutevedzana kwemasvomhu, chikamu chechishanu chiri 20 uye chikamu chechisere chiri 35. Sarudza chikamu chekutanga \(a\) uye musiyano wakajairika \(d\) wechikamu.
Kukurukurirana:
Kubva pamubvunzo, tinoziva:
\[ U_5 = a + 4d = 20 \]
\[ U_8 = a + 7d = 35 \]
Bvisa ma equation ese ari maviri kuti ubvise \(a\):
\[ (a + 7d) – (a + 4d) = 35 – 20 \]
\[3d = 15 \]
\[d = 5 \]
Iye zvino tava kutsiva \(d = 5\) kuti tiwane \(a\):
\[ a + 4 \cdot 5 = 20 \]
\[a + 20 = 20 \]
\[ a = 0 \]
Saka, izwi rekutanga remutsara uyu ndi0 uye musiyano ndi5.
Muenzaniso Mubvunzo 3
Mubvunzo:
Chii chinosanganisira mazwi makumi maviri ekutanga echikamu chekuverenga masvomhu chine izwi rekutanga riri \(a = 2\) uye musiyano wakafanana \(d = 4\)?
Kukurukurirana:
Huwandu hwemashoko ekutanga e n e arithmetic sequence hunogona kuverengerwa uchishandisa fomura:
\[ S_n = \frac{n}{2} \left( 2a + (n-1)d \right) \]
Kune uyu mutsetse:
\[ S_{20} = \frac{20}{2} \left( 2 \cdot 2 + (20-1) \cdot 4 \right) \]
\[ S_{20} = 10 \kuruboshwe( 4 + 76 \kurudyi) \]
\[ S_{20} = 10 \cdot 80 \]
\[ S_{20} = 800 \]
Saka, huwandu hwemashoko makumi maviri ekutanga emutsara uyu i800.
Muenzaniso Mubvunzo 4
Mubvunzo:
Temu yechitatu yekutevera kwemasvomhu i15 uye temu yechinomwe i27. Tsvaga temu yechi12 yekutevera.
Kukurukurirana:
Kutanga, tinofanira kuwana kukosha \(a\) uye \(d\). Kubva pamubvunzo, tinoziva:
\[ U_3 = a + 2d = 15 \]
\[ U_7 = a + 6d = 27 \]
Bvisa ma equation ese ari maviri kuti ubvise \(a\):
\[ (a + 6d) – (a + 2d) = 27 – 15 \]
\[4d = 12 \]
\[d = 3 \]
Iye zvino tava kutsiva \(d = 3\) kuti tiwane \(a\):
\[ a + 2 \cdot 3 = 15 \]
\[a + 6 = 15 \]
\[ a = 9 \]
Kushandisa fomura yenth term kuti uwane term yechi12:
\[ U_{12} = a + 11d \]
\[ U_{12} = 9 + 11 \cdot 3 \]
\[ U_{12} = 9 + 33 \]
\[ U_{12} = 42 \]
Saka, chikamu chechishanu chemutsara uyu i42.
Muenzaniso Mubvunzo 5
Mubvunzo:
Kutevedzana kwemasvomhu netemu yekutanga \(a\) nemusiyano wakajairika \(d\) kune huwandu hwemashoko gumi ekutanga, kureva 55. Kana \(d = 1\), tsvaga temu yekutanga \(a\).
Kukurukurirana:
Zvapiwa \(d = 1\) uye \(S_{10} = 55\). Shandisa fomura yehuwandu hwemashoko ekutanga:
\[ S_n = \frac{n}{2} (2a + (n-1)d) \]
Kune n = 10:
\[ S_{10} = \frac{10}{2} (2a + 9 \cdot 1) = 55 \]
\[ 5 (2a + 9) = 55 \]
\[ 2a + 9 = 11 \]
\[ 2a = 2 \]
\[ a = 1 \]
Saka, izwi rekutanga remutsara uyu ndi1.
Mhedziso
Kutevedzana kwemasvomhu ipfungwa huru mumasvomhu inobatsira zvikuru muminda yakasiyana-siyana. Muchinyorwa chino, takurukura mienzaniso yakati wandei yemasvomhu nemhinduro dzawo. Kunzwisisa kwakanaka kwemafomula uye hunhu hwekutanga hwemasvomhu kuchabatsira zvikuru mukugadzirisa matambudziko akasiyana-siyana ane chekuita nenyaya iyi.
Nekudzidzira mienzaniso yakaita seyiyi iri pamusoro, zvinotarisirwa kuti uchava neunyanzvi uye nekukurumidza mukugadzirisa matambudziko ane chekuita nekutevedzana kwemasvomhu.
