Su'aalo Tusaale ah oo Ka Hadlaya Taxanaha Xisaabta
Taxanaha xisaabtu waa fikrad aasaasi ah oo ku jirta xisaabta oo inta badan ka soo baxda dhibaatooyin kala duwan, labadaba dugsiga sare iyo waxbarashada sare. Fikraddani waxay ku lug leedahay taxan tirooyin ah halkaas oo erey kastaa uu ka dhashay ku darista ama ka jarista tiro joogto ah oo ka timid ereygii hore. Maqaalkan, waxaan kaga hadli doonnaa dhowr tusaale oo dhibaatooyin ah iyo xalalkooda si loo sii fahmo fikradda taxanaha xisaabta.
Fahmidda Taxanaha Xisaabta
Taxane xisaabeed waa taxane leh farqi joogto ah (farqi) oo u dhexeeya laba erey oo isku xiga. Tusaale ahaan, haddii taxanaha xisaabeed uu leeyahay ereyga koowaad \(a\) iyo farqiga \(d\), markaa ereyada waxaa loo qori karaa sidan soo socota:
\[ a, a + d, a + 2d, a + 3d, \ldots \]
Haddii aan rabno inaan helno ereyga naad ee taxanahan, qaacidada ereyga naad (\(U_n\)) waa:
\[ U_n = a + (n-1)d \]
Dhanka kale, wadarta ereyada n ee ugu horreeya ee taxanaha xisaabta (\(S_n\)) waxaa lagu xisaabin karaa qaacidada:
\[ S_n = \frac{n}{2} (2a + (n-1)d) \]
Su'aalo iyo Doodo Tusaale ah
Su'aal Tusaale 1aad
Su'aal: Iyadoo la bixinayo taxane xisaabeed oo leh ereyga koowaad \(a = 5\) iyo farqiga guud \(d = 3\). Soo hel ereyga 10aad ee taxanaha.
Dood:
Si loo helo ereyga 10aad (\(U_{10}\)), waxaan isticmaali karnaa qaacidada ereyga naad:
\[ U_{10} = a + (10-1)d \]
\[ U_{10} = 5 + (9 \cdot 3) \]
\[ U_{10} = 5 + 27 \]
\[ U_{10} = 32 \]
Markaa, 10-ka jeer ee taxanaha waa 32.
Su'aal Tusaale 2aad
Su'aal: Soo hel wadarta 15-ka erey ee ugu horreeya taxanaha xisaabta oo ereygooda koowaad uu yahay \(a = 4\) farqiga guudna uu yahay \(d = 7\).
Dood:
Si loo helo wadarta 15-ka erey ee ugu horreeya (\(S_{15}\)), waxaan u isticmaali karnaa qaacidada wadarta ereyada n ee ugu horreeya:
\[ S_{15} = \frac{15}{2} (2a + (15-1)d) \]
\[ S_{15} = \frac{15}{2} (2 \cdot 4 + 14 \cdot 7) \]
\[ S_{15} = \frac{15}{2} (8 + 98) \]
\[ S_{15} = \frac{15}{2} \cdot 106 \]
\[ S_{15} = 15 \cdot 53 \]
\[ S_{15} = 795 \]
Markaa, wadarta 15-ka erey ee ugu horreeya taxanaha waa 795.
Su'aal Tusaale 3aad
Su'aal: Waa la ogyahay in ereyga 5aad ee taxanaha xisaabta uu yahay 20, ereyga 12aadna uu yahay 48. Soo hel ereyga koowaad (\(a\)) iyo farqiga guud (\(d\)) ee taxanaha.
Dood:
Laga soo bilaabo shuruudaha la bixiyay:
\[ U_5 = a + 4d = 20 \]
\[ U_{12} = a + 11d = 48 \]
Waxaan haynaa laba isle'eg oo toosan oo leh laba doorsoome oo aan xallin karno:
1. \( a + 4d = 20 \)
2. \( a + 11d = 48 \)
Laga soo bilaabo isla'egta 1, waxaan ku qeexi karnaa \(a\) iyadoo la adeegsanayo \(d\):
\[ a = 20 – 4d \]
Hadda waxaan \(a\) ku beddeleynaa isle'egta 2:
\[ 20 – 4d + 11d = 48 \]
\[ 20 + 7d = 48 \]
\[ 7d = 28 \]
\[ d = 4 \]
Hadda waxaan qiimaha \(d\) ku beddeleynaa isla'egta \(a = 20 - 4d\):
\[ a = 20 – 4 \cdot 4 \]
\[ a = 20 - 16 \]
\[ a = 4 \]
Markaa, ereyga koowaad ee taxanaha waa 4 farqiga guudna waa 4.
Su'aal Tusaale 4aad
Su'aal: Immisa erey ayaa loo baahan yahay taxanaha xisaabta oo leh ereyga koowaad \(a = 2\) iyo farqiga guud \(d = 5\) si loo wadar ahaan u noqdo 200?
Dood:
Xaaladdan, waxaan u baahanahay inaan helno wadarta ereyada n ee ugu horreeya (\(S_n\)) oo la mid ah 200. Isticmaal qaacidada wadarta ereyada n ee ugu horreeya:
\[ S_n = \frac{n}{2} (2a + (n-1)d) = 200 \]
Ku beddel qiimaha \(a\) iyo \(d\):
\[ \frac{n}{2} (2 \cdot 2 + (n-1) \cdot 5) = 200 \]
\[ \ frac {n}{2} (4 + 5n - 5) = 200 \]
\[ \ frac {n}{2} (5n - 1) = 200 \]
\[ n (5n – 1) = 400 \]
Kani waa isle'eg labajibbaaran. Si aan u xallino, waxaan beddeleynaa qaabkeeda:
\[ 5n^2 – n – 400 = 0 \]
Isticmaal qaacidada labajibbaaran \(ax^2 + bx + c = 0\):
\[ n = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Xaaladdan \(a = 5\), \(b = -1\), iyo \(c = -400\):
\[ n = \frac{-(-1) \pm \sqrt{(-1)^2 – 4 \cdot 5 \cdot (-400)}}{2 \cdot 5} \]
\[ n = \frac{1 \pm \sqrt{1 + 8000}}{10} \]
\[ n = \frac{1 \pm \sqrt{8001}}{10} \]
Qiimaha \(\sqrt{8001}\) wuxuu ku dhow yahay 89.42, markaa:
\[ n = \frac{1 \pm 89.42}{10} \]
Waxaan qaadanaa qiimayaasha togan:
\[ n = \ frac{1 + 89.42}{10} \]
\[ n \qiyaastii \frac{90.42}{10} \]
\[ n \qiyaastii 9.042 \]
Markaa, tirada ereyada loo baahan yahay waa 9 erey (haddii la soo koobay).
Gabagabo
Taxanaha xisaabtu waa mawduuc muhiim u ah xisaabta. Faham qoto dheer oo ku saabsan ereyga koowaad, farqiga guud, ereyga naad, iyo wadarta ereyada n ee ugu horreeya waa mid qiimo leh oo lagu xallinayo dhibaatooyin kala duwan. Iyadoo la adeegsanayo tusaalooyinka iyo doodaha kor ku xusan, waxaa la rajeynayaa in akhristayaashu ay si fiican u fahmi doonaan fikradaha aasaasiga ah ee taxanaha xisaabta.
