Imibuzo Eyisibonelo Exoxa Ngezilandelaniso Zezibalo
Ukulandelana kwezibalo kuwumqondo oyisisekelo ezibalweni ovame ukuvela ezivivinyweni ezahlukene kanye nezicelo zomhlaba wangempela. Ukulandelana kwezibalo kuwukulandelana kwezinombolo okunomehluko oqhubekayo phakathi kwamagama amabili alandelanayo. Kulesi sihloko, sizohlola umqondo wokulandelana kwezibalo ngokujulile ngezinkinga eziningana zezibonelo ezihambisana nezincazelo eziphelele.
Izincazelo kanye nemibhalo
Ngaphambi kokuthi singene ezinkingeni zesibonelo, kubalulekile ukuqonda ukuthi inothi livame ukusetshenziswa kanjani ku-arithmetic sequences. Uma i-\(a\) iyitemu lokuqala kanye ne-\(d\) ingumehluko ovamile (umehluko oqhubekayo), khona-ke i-arithmetic sequence ingabhalwa kanje:
\[ a, a+d, a+2d, a+3d, \ldots, a+(n-1)d \]
Igama elithi nth (Un) lalolu chungechunge lingakhiwa kanje:
\[ U_n = a + (n-1)d \]
Okulandelayo yimibuzo eminingana eyisibonelo kanye nezingxoxo zayo ukuze kube lula ukuqonda ukulandelana kwezibalo.
Isibonelo Umbuzo 1
Umbuzo:
Uma unikezwe ukulandelana kwezibalo ngetemu lokuqala \(a = 5\) kanye nomehluko ojwayelekile \(d = 3\). Thola ithemu yeshumi yokulandelana.
Ingxoxo:
Ukusebenzisa ifomula ejwayelekile yethemu le-nth, okungukuthi \( U_n = a + (n-1)d \):
\[ U_{10} = 5 + (10-1) \cdot 3 \]
\[ U_{10} = 5 + 9 \cdot 3 \]
\[ U_{10} = 5 + 27 \]
\[ U_{10} = 32 \]
Ngakho-ke, itemu lesi-10 lochungechunge lingu-32.
Isibonelo Umbuzo 2
Umbuzo:
Uma unikezwe ukulandelana kwezibalo, ithemu lesihlanu lingu-20 kanti ithemu lesishiyagalombili lingu-35. Thola ithemu lokuqala \(a\) kanye nomehluko ojwayelekile \(d\) wokulandelana.
Ingxoxo:
Kusukela embuzweni, siyazi:
\[ U_5 = a + 4d = 20 \]
\[ U_8 = a + 7d = 35 \]
Susa zombili izilinganiso ukuze ususe \(a\):
\[ (a + 7d) – (a + 4d) = 35 – 20 \]
\[ 3d = 15 \]
\[d = 5 \]
Manje sifaka esikhundleni se-\(d = 5\) ukuthola i-\(a\):
\[ a + 4 \cdot 5 = 20 \]
\[a + 20 = 20 \]
\[ a = 0 \]
Ngakho-ke, igama lokuqala lochungechunge lingu-0 kanti umehluko ungu-5.
Isibonelo Umbuzo 3
Umbuzo:
Iyini ingqikithi yamagama okuqala angu-20 ochungechunge lwezibalo olunegama lokuqala elingu-\(a = 2\) kanye nomehluko ovamile \(d = 4\)?
Ingxoxo:
Isamba samagama okuqala ka-n ochungechungeni lwezibalo singabalwa kusetshenziswa ifomula:
\[ S_n = \frac{n}{2} \left( 2a + (n-1)d \right) \]
Kulo mugqa:
\[ S_{20} = \frac{20}{2} \left( 2 \cdot 2 + (20-1) \cdot 4 \right) \]
\[ S_{20} = 10 \kwesobunxele( 4 + 76 \kwesokudla) \]
\[ S_{20} = 10 \cdot 80 \]
\[ S_{20} = 800 \]
Ngakho-ke, isamba samagama okuqala angu-20 ochungechunge singama-800.
Isibonelo Umbuzo 4
Umbuzo:
Ithemu lesithathu lochungechunge lwezibalo lingu-15 kanti ithemu lesikhombisa lingu-27. Thola ithemu le-12 lochungechunge.
Ingxoxo:
Okokuqala, sidinga ukuthola amanani \(a\) kanye \(d\). Kusukela embuzweni, siyazi:
\[ U_3 = a + 2d = 15 \]
\[ U_7 = a + 6d = 27 \]
Susa zombili izilinganiso ukuze ususe \(a\):
\[ (a + 6d) – (a + 2d) = 27 – 15 \]
\[ 4d = 12 \]
\[d = 3 \]
Manje sifaka esikhundleni se-\(d = 3\) ukuthola i-\(a\):
\[ a + 2 \cdot 3 = 15 \]
\[a + 6 = 15 \]
\[ a = 9 \]
Ukusebenzisa ifomula yethemu le-nth ukuthola ithemu le-12:
\[ U_{12} = a + 11d \]
\[ U_{12} = 9 + 11 \cdot 3 \]
\[ U_{12} = 9 + 33 \]
\[ U_{12} = 42 \]
Ngakho-ke, itemu lesi-12 lochungechunge lingu-42.
Isibonelo Umbuzo 5
Umbuzo:
Uchungechunge lwezibalo olunegama lokuqala \(a\) kanye nomehluko ojwayelekile \(d\) lunesamba samagama okuqala ayi-10, okungukuthi angu-55. Uma \(d = 1\), nquma igama lokuqala \(a\).
Ingxoxo:
Kunikezwe \(d = 1\) kanye \(S_{10} = 55\). Sebenzisa ifomula yesamba samagama okuqala:
\[ S_n = \frac{n}{2} (2a + (n-1)d) \]
Ku-n = 10:
\[ S_{10} = \frac{10}{2} (2a + 9 \cdot 1) = 55 \]
\[ 5 (2a + 9) = 55 \]
\[ 2a + 9 = 11 \]
\[ 2a = 2 \]
\[ a = 1 \]
Ngakho-ke, igama lokuqala lochungechunge lingu-1.
Isiphetho
Ukulandelana kwezibalo kuwumqondo oyisisekelo kwizibalo owusizo kakhulu emikhakheni eyahlukene. Kulesi sihloko, sixoxe ngezibonelo eziningana zokulandelana kwezibalo kanye nezixazululo zazo. Ukuqonda kahle amafomula kanye nezakhiwo eziyisisekelo zokulandelana kwezibalo kuzosiza kakhulu ekuxazululeni izinhlobo ezahlukene zezinkinga ezihlobene nalesi sihloko.
Ngokuzijwayeza izibonelo ezifana nalezi ezingenhla, kunethemba lokuthi uzoba nekhono futhi usheshe ekuxazululeni izinkinga ezihilela ukulandelana kwezibalo.
