Zitsanzo za Mafunso Okhudza Mndandanda wa Zoyimira Zosatha
Mndandanda wa geometric wopanda malire ndi mndandanda womwe umaphatikizapo mawu osatha mu kupita patsogolo kwa geometric. Mndandanda uwu uli ndi zofunikira zenizeni kuti chiwerengero chawo (kapena malire) chiwerengedwe. M'nkhaniyi, tikambirana za lingaliro loyambira la mndandanda wa geometric wopanda malire, zofunikira pakuzipanga, ndi zitsanzo zingapo za mavuto ndi mayankho awo.
Malingaliro Oyambira a Mndandanda Wosatha wa Jiyometri
Kwenikweni, mndandanda wa manambala a geometric ndi mndandanda wa manambala momwe mawu aliwonse pambuyo pa oyamba amapezeka pochulukitsa mawu am'mbuyomu ndi chosasintha chotchedwa common ratio (r). Tiyerekeze kuti tili ndi mndandanda wa manambala a geometric:
\[ a, ar, ar^2, ar^3, ar^4, \ldots \]
Pa mndandanda wopanda malire wa geometric, timaganizira kuchuluka kwa mawu onse mu mndandanda. Chiwerengero cha mndandanda uwu chimatanthauzidwa motere:
\[ S = a + ar + ar^2 + ar^3 + ar^4 + \ldots \]
Chiwerengero cha mndandanda wopanda malire wa geometric chimalumikizana (chili ndi chiwerengero chotsimikizika) ngati ndipo kokha ngati chiŵerengero \( |r| < 1 \). Ngati \( |r| \geq 1 \), ndiye kuti mndandandawo umasiyana ndipo ulibe chiwerengero chotsimikizika (chimapita ku chosatha).
Ngati \( |r| < 1 \), chiwerengero cha S cha mndandanda wa geometric wopanda malire chingatanthauzidwe motere: \[ S = \frac{a}{1-r} \] pomwe: - \( S \) ndi chiwerengero cha mndandanda, - \( a \) ndi gawo loyamba, - \( r \) ndi chiwerengero. Zitsanzo za Mafunso ndi Kukambirana Chitsanzo Funso 1: Pezani chiwerengero cha mndandanda wa geometric wopanda malire wa mndandanda wotsatira: \[ 3 + 1.5 + 0.75 + 0.375 + \ldots \] Kukambirana: Tiyeni tidziwe zinthu zofunika kwambiri za mndandanda: Gawo loyamba \( a = 3 \) Chiŵerengero \( r \) chingapezeke pogawa gawo lachiwiri ndi gawo loyamba, ndiko kuti: \[ r = \frac{1.5}{3} = 0.5 \] Popeza \( |r| = 0.5 < 1 \), mndandanda uwu umalumikizana ndipo tikhoza kuwerengera chiwerengero cha mndandanda wopanda malire. Gwiritsani ntchito njira yowerengera kuchuluka kwa mndandanda wa geometric wopanda malire: \[ S = \frac{a}{1-r} \] \[ S = \frac{3}{1-0.5} \] \[ S = \frac{3}{0.5} \] \[ S = 6 \] Chifukwa chake, kuchuluka kwa mndandanda wa geometric wopanda malire ndi 6. Chitsanzo Funso 2 Funso: Pezani kuchuluka kwa mndandanda wa geometric wopanda malire ndi mawu oyamba 8 ndi chiŵerengero \( r = -\frac{1}{3} \). Kukambirana: Gawo loyamba \( a = 8 \) Chiŵerengero \( r = -\frac{1}{3} \) Popeza \( |r| = \frac{1}{3} < 1 \), mndandanda uwu umalumikizana ndipo tikhoza kuwerengera kuchuluka kwa mndandanda wopanda malire. Gwiritsani ntchito fomula ya kuchuluka kwa mndandanda wopanda malire wa geometric: \[ S = \frac{a}{1-r} \] \[ S = \frac{8}{1 - \left(-\frac{1}{3}\right)} \] \[ S = \frac{8}{1 + \frac{1}{3}} \] \[ S = \frac{8}{\frac{4}{3}} \] \[ S = 8 \times \frac{3}{4} \] \[ S = 6 \] Chifukwa chake, kuchuluka kwa mndandanda wopanda malire wa geometric ndi 6. Chitsanzo Funso 3 Funso: Kodi mndandanda wotsatira uli ndi kuchuluka kosatha? Ngati ndi choncho, pezani kuchuluka. \[ 5 + 2.5 + 1.25 + 0.625 + \ldots \] Kukambirana: Gawo loyamba \( a = 5 \) Chiŵerengero \( r \) chingapezeke pogawa gawo lachiwiri ndi gawo loyamba, ndiko kuti: \[ r = \frac{2.5}{5} = 0.5 \] Popeza \( |r| = 0.5 < 1 \), mndandanda uwu umalumikizana ndipo tikhoza kuwerengera kuchuluka kwa mndandanda wopanda malire. Gwiritsani ntchito fomula ya kuchuluka kwa mndandanda wopanda malire wa geometric: \[ S = \frac{a}{1-r} \] \[ S = \frac{5}{1-0.5} \] \[ S = \frac{5}{0.5} \] \[ S = 10 \] Chifukwa chake, kuchuluka kwa mndandanda wopanda malire wa geometric ndi 10. Chitsanzo Funso 4 Funso: Dziwani ngati mndandanda wotsatirawu ndi wogwirizana kapena wosiyana: \[ 4 - 6 + 9 - 13.5 + \ldots \] Kukambirana: Gawo loyamba \( a = 4 \) Chiŵerengero \( r \) chingapezeke pogawa gawo lachiwiri ndi gawo loyamba, ndiko kuti: \[ r = \frac{-6}{4} = -1.5 \] Popeza \( |r| = 1.5 > 1 \), mndandanda uwu ndi wosiyana ndipo ulibe chiwerengero chotsimikizika.Kotero, mndandandawu ndi wosiyana.
Chitsanzo cha Funso 5
Funso: Tiyerekeze kuti muli ndi mndandanda wopanda malire wotsatira:
\[ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots \]
Dziwani kuchuluka kwa mndandanda.
Kukambirana:
Gawo loyamba \( a = \frac{1}{2} \)
Chiŵerengero \(r \) chingapezeke pogawa gawo lachiwiri ndi gawo loyamba, ndiko kuti:
\[ r = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{2} \]
Popeza \( |r| = \frac{1}{2} < 1 \), mndandanda uwu umalumikizana ndipo tikhoza kuwerengera kuchuluka kwa mndandanda wopanda malire. Gwiritsani ntchito fomula ya kuchuluka kwa mndandanda wopanda malire wa geometric: \[ S = \frac{a}{1-r} \] \[ S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} \] \[ S = \frac{\frac{1}{2}}{\frac{1}{2}} \] \[ S = 1 \] Chifukwa chake, kuchuluka kwa mndandanda wopanda malire wa geometric ndi 1. Mapeto Mndandanda wopanda malire wa geometric ndi lingaliro lofunika la masamu lomwe limagwiritsidwa ntchito kwambiri m'magawo osiyanasiyana. Kuti tidziwe kuchuluka kwa mndandanda wopanda malire wa geometric, tiyenera kuwonetsetsa kuti chiŵerengero \( |r| < 1 \). Chifukwa chake, kuchuluka kwa mndandanda kumatha kuwerengedwa pogwiritsa ntchito njira yosavuta komanso yolunjika. Kuchokera ku zitsanzo zamavuto pamwambapa, titha kuwona kuti njira iyi imapangitsa kuti zikhale zosavuta kwambiri kuthetsa mavuto okhudzana ndi mndandanda wopanda malire wa geometric.
