Imibuzo Eyisibonelo Exoxa Ngochungechunge Lwezibalo
Ukulandelana kwezibalo kuwumqondo oyisisekelo ezibalweni ovame ukuvela ezinkingeni ezahlukene, kokubili esikoleni samabanga aphezulu kanye nasemfundweni ephakeme. Lo mqondo uhilela ukulandelana kwezinombolo lapho ithemu ngalinye liwumphumela wokwengeza noma ukususa inombolo engaguquki ethemini elidlule. Kulesi sihloko, sizoxoxa ngezinkinga eziningana eziyizibonelo kanye nezixazululo zazo ukuze siqonde kabanzi umqondo wokulandelana kwezibalo.
Ukuqonda Uchungechunge Lwezibalo
Uchungechunge lwezibalo luchungechunge olunomehluko ongaguquki (umehluko) phakathi kwamagama amabili alandelanayo. Isibonelo, uma uchungechunge lwezibalo lunegama lokuqala \(a\) kanye nomehluko \(d\), khona-ke amagama angabhalwa kanje:
\[ a, a + d, a + 2d, a + 3d, \ldots \]
Uma sifuna ukuthola ithemu ye-nth yalolu chungechunge, ifomula yethemu ye-nth (\(U_n\)) ithi:
\[ U_n = a + (n-1)d \]
Okwamanje, isamba samagama okuqala ka-n ochungechungeni lwezibalo (\(S_n\)) singabalwa kusetshenziswa ifomula:
\[ S_n = \frac{n}{2} (2a + (n-1)d) \]
Imibuzo Eyisibonelo Nengxoxo
Isibonelo Umbuzo 1
Umbuzo: Uma unikezwe uchungechunge lwezibalo olunegama lokuqala \(a = 5\) kanye nomehluko ojwayelekile \(d = 3\). Thola itemu leshumi lochungechunge.
Ingxoxo:
Ukuze sithole ithemu yeshumi (\(U_{10}\)), singasebenzisa ifomula yethemu ye-nth:
\[ U_{10} = a + (10-1)d \]
\[ U_{10} = 5 + (9 \cdot 3) \]
\[ U_{10} = 5 + 27 \]
\[ U_{10} = 32 \]
Ngakho-ke, itemu le-10 lochungechunge lingu-32.
Isibonelo Umbuzo 2
Umbuzo: Thola isamba samagama okuqala angu-15 ochungechunge lwezibalo lapho igama lokuqala lingu-\(a = 4\) kanye nomehluko ovamile ungu-\(d = 7\).
Ingxoxo:
Ukuze sithole isamba samagama okuqala angu-15 (\(S_{15}\)), singasebenzisa ifomula yesamba samagama okuqala angu-n:
\[ S_{15} = \frac{15}{2} (2a + (15-1)d) \]
\[ S_{15} = \frac{15}{2} (2 \cdot 4 + 14 \cdot 7) \]
\[ S_{15} = \frac{15}{2} (8 + 98) \]
\[ S_{15} = \frac{15}{2} \cdot 106 \]
\[ S_{15} = 15 \cdot 53 \]
\[ S_{15} = 795 \]
Ngakho-ke, isamba semigomo yokuqala engu-15 yochungechunge singama-795.
Isibonelo Umbuzo 3
Umbuzo: Kuyaziwa ukuthi ithemu yesi-5 yochungechunge lwezibalo ingu-20 kanti ithemu yesi-12 ingu-48. Thola ithemu yokuqala (\(a\)) kanye nomehluko ojwayelekile (\(d\)) wochungechunge.
Ingxoxo:
Kusukela ezimweni ezinikeziwe:
\[ U_5 = a + 4d = 20 \]
\[ U_{12} = a + 11d = 48 \]
Sinezibalo ezimbili eziqondile ezineziguquguquko ezimbili esingazixazulula:
1. \( a + 4d = 20 \)
2. \( a + 11d = 48 \)
Kusukela ku-equation 1, singaveza \(a\) ngokwe-\(d\):
\[a = 20 – 4d \]
Manje sifaka i-\(a\) ku-equation 2:
\[ 20 – 4d + 11d = 48 \]
\[ 20 + 7d = 48 \]
\[ 7d = 28 \]
\[d = 4 \]
Manje sifaka inani lika-\(d\) esilinganisweni \(a = 20 – 4d\):
\[ a = 20 – 4 \cdot 4 \]
\[a = 20 – 16 \]
\[ a = 4 \]
Ngakho-ke, itemu lokuqala lochungechunge lingu-4 kanti umehluko ojwayelekile ungu-4.
Isibonelo Umbuzo 4
Umbuzo: Mangaki amagama adingekayo ochungechungeni lwezibalo olunethemu yokuqala \(a = 2\) kanye nomehluko ojwayelekile \(d = 5\) ukuze kuhlanganiswe ku-200?
Ingxoxo:
Kulesi simo, sidinga ukuthola isamba samagama okuqala ka-n (\(S_n\)) esilingana no-200. Sebenzisa ifomula yesamba samagama okuqala ka-n:
\[ S_n = \frac{n}{2} (2a + (n-1)d) = 200 \]
Faka amanani esikhundleni sika-\(a\) kanye no-\(d\):
\[ \frac{n}{2} (2 \cdot 2 + (n-1) \cdot 5) = 200 \]
\[ \frac{n}{2} (4 + 5n – 5) = 200 \]
\[ \frac{n}{2} (5n – 1) = 200 \]
\[ n (5n – 1) = 400 \]
Lesi yisibalo esiphindwe kane. Ukuze sisixazulule, sishintsha isimo saso:
\[ 5n^2 – n – 400 = 0 \]
Sebenzisa ifomula ye-quadratic \(ax^2 + bx + c = 0\):
\[ n = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Kulesi simo \(a = 5\), \(b = -1\), kanye \(c = -400\):
\[ n = \frac{-(-1) \pm \sqrt{(-1)^2 – 4 \cdot 5 \cdot (-400)}}{2 \cdot 5} \]
\[ n = \frac{1 \pm \sqrt{1 + 8000}}{10} \]
\[ n = \frac{1 \pm \sqrt{8001}}{10} \]
Inani lika-\(\sqrt{8001}\) liseduze no-89.42, bese kuba:
\[ n = \frac{1 \pm 89.42}{10} \]
Sithatha izindinganiso ezinhle:
\[ n = \frac{1 + 89.42}{10} \]
\[ n \cishe \frac{90.42}{10} \]
\[ n \cishe 9.042 \]
Ngakho-ke, inani lamagama adingekayo yi-9 (uma ehlanganisiwe).
Isiphetho
Ukulandelana kwezibalo kuyisihloko esibalulekile kwizibalo. Ukuqonda kahle ithemu lokuqala, umehluko ovamile, ithemu le-n, kanye nesamba samagama okuqala e-n kubaluleke kakhulu ekuxazululeni izinkinga ezahlukahlukene. Ngokusebenzisa izibonelo nezingxoxo ezingenhla, kunethemba lokuthi abafundi bazothola ukuqonda okungcono kwemiqondo eyisisekelo yokulandelana kwezibalo.
