Podaj brakujące elementy dla ciągu geometrycznego a) a1=2, an=1458 Sn=2186 Oblicz q i n. b) a1 = [latex] -sqrt{3} [/latex] q=[latex] sqrt{6} [/latex] an=[latex] -108sqrt{2} [/latex]. Oblicz n i Sn.

a) an=a1 * q^(n-1) 1458=2*q^(n-1) q^(n-1)=729 [latex]Sn= a_1frac{1-q^n}{1-q}= extgreater 2186=2* frac{1-729*q}{1-q}= extgreater 1093= frac{1-729q}{1-q}= extgreater [/latex] [latex]1093-1093q=1-729q = extgreater -364q=-1092= extgreater q=3[/latex] 3^(n-1)=729 n-1=6 n=7 b) an=a1*q^(n-1) [latex]-108 sqrt{2}=- sqrt{3}* sqrt{6}^{n-1}= extgreater frac{108 sqrt{2} }{ sqrt{3} }= sqrt{6}^{n-1}= extgreater frac{108 sqrt{6} }{ 3 }= sqrt{6}^{n-1}= extgreater [/latex] [latex] 36 sqrt{6}= sqrt{6}^{n-1}= extgreater 6^2*6^{ frac{1}{2} }=6^{ frac{1}{2}(n-1) }= extgreater 6^{ frac{5}{2} }=6^{ frac{1}{2}(n-1) }= extgreater [/latex] [latex]{ frac{5}{2}= frac{1}{2}(n-1) = extgreater { frac{5}{2}= frac{1}{2}n- frac{1}{2}= extgreater frac{1}{2}n= frac{6}{2}= extgreater n=6 [/latex] Sn=S6=[latex]- sqrt{3} frac{1- sqrt{6} ^6}{1- sqrt{6} }= - sqrt{3} frac{1- 216}{1- sqrt{6} }* frac{1+ sqrt{6} }{1+ sqrt{6} }=215sqrt{3}* frac{1+ sqrt{6} }{1-6}= frac{215 sqrt{3} }{-5}(1+ sqrt{6} )=[/latex] [latex] frac{215 sqrt{3} }{-5}(1+ sqrt{6} )=- 43 sqrt{3}(1+ sqrt{6}) [/latex]