Tabelka - patrz załącznik n-ty wyraz ciągu geometrycznego o wyrazie pierwszym a₁ i ilorazie q, wyraża się wzorem: [latex]a_n = a_1 cdot q^{n - 1}[/latex] Suma n - początkowych wyrazów ciągu geometrycznego o wyrazie pierwszym a₁ i ilorazie q, wyraża się wzorem: [latex]S_n = a_1 cdot frac{1 - q^n}{1 - q}[/latex] 1. [latex]a_1 = 2; q = 3; S_n = 242 \\ S_n = a_1 cdot frac{1 - q^n}{1 - q} \ 2 cdot frac{1 - 3^n}{1 - 3} = 242 \ 2 cdot frac{1 - 3^n}{-2} = 242 \ - (1 - 3^n) = 242 \ - 1 + 3^n = 242 \ 3^n = 242 + 1 \ 3^n = 243 \ 3^n = 3^5 \ n = 5 \\ a_n = a_1 cdot q^{n - 1} \ a_5 = 2 cdot 3^{5 - 1} = 2 cdot 3^4 = 2 cdot 81 = 162[/latex] 2. [latex]q = frac{1}{2}; n = 7; a_n = a_7 = frac{1}{2}\\ a_n = a_1 cdot q^{n - 1}\ a_1 cdot (frac{1}{2})^{7 - 1} = frac{1}{2} \ a_1 cdot (frac{1}{2})^6 = frac{1}{2} \ a_1 cdot frac{1}{64} = frac{1}{2} / cdot 64 \ a_1 = 32 \\ S_n = a_1 cdot frac{1 - q^n}{1 - q} \ S_7 = 32 cdot frac{1 - (frac{1}{2})^7}{1 - frac{1}{2}} = 32 cdot frac{1 - frac{1}{128}}{frac{1}{2}} = 32 cdot frac{127}{128} cdot 2 = frac{127}{2} = 63frac{1}{2}[/latex] 3. [latex]q = 3; n = 10; S_n = S_{10}= 1023 \\ S_n = a_1 cdot frac{1 - q^n}{1 - q} \ a_1 cdot frac{1 - 3^{10}}{1 - 3}= 1023 \ a_1 cdot frac{1 - 59049}{-2}= 1023 \ a_1 cdot frac{- 59048}{-2}= 1023 \ a_1 cdot 29524 = 1023 / : 29524 \ a_1 = frac{1023}{29524} \ a_1 = frac{93}{2684}\\ a_n = a_1 cdot q^{n - 1} \ a_{10} = frac{93}{2684}cdot 3^{10 - 1}= frac{93}{2684}cdot 19683 = frac{1830519}{2684}= 682frac{31}{2684}[/latex] 4. [latex]a_1 = frac{1}{25}; n = 7; a_n = a_7 = 625 \\ a_n = a_1 cdot q^{n - 1} \ frac{1}{25}cdot q^{7 - 1} = 625 / cdot 25 \ q^6 = 625 cdot 25 \ q^6 = 5^4 cdot 5^2 \ q^6 = 5^6 \ q = 5 \\ S_n = a_1 cdot frac{1 - q^n}{1 - q} \ S_7 = frac{1}{25}cdot frac{1 - 5^7}{1 - 5} = frac{1}{25}cdot frac{1 - 78125}{-4}= frac{1}{25}cdot frac{- 78124}{-4}=frac{1}{25}cdot 19683 = frac{19683}{25}= \ = 781frac{6}{25}[/latex] 5. [latex]a_1 = sqrt{8} = sqrt{4 cdot 2}= 2sqrt{2}; q = -sqrt{2}; n = 8 \\ a_n = a_1 cdot q^{n - 1} \ a_8 = 2sqrt{2} cdot (-sqrt{2})^{8 - 1} = 2sqrt{2} cdot (-sqrt{2})^7 = 2sqrt{2} cdot (-sqrt{2^7}) = \ = 2sqrt{2} cdot (-sqrt{2^6 cdot 2}) = 2sqrt{2} cdot (-2^3sqrt{2}) = 2sqrt{2} cdot (-8sqrt{2}) = - 32[/latex] [latex]S_n = a_1 cdot frac{1 - q^n}{1 - q} \ S_8 = 2sqrt{2}cdot frac{1 - (-sqrt{2})^8}{1 - (-sqrt{2})}= 2sqrt{2}cdot frac{1 - sqrt{2^8}}{1 + sqrt{2}}= 2sqrt{2}cdot frac{1 - 2^4}{1 + sqrt{2}}= 2sqrt{2}cdot frac{1 - 16}{1 + sqrt{2}}=\ =2sqrt{2}cdot frac{-15}{1 + sqrt{2}}= frac{-30sqrt{2}}{1 + sqrt{2}}= frac{-30sqrt{2}cdot (1 - sqrt{2})}{(1 + sqrt{2})(1 - sqrt{2})}= frac{-30sqrt{2}+60}{1-2}= frac{-30sqrt{2}+60}{-1}=\ =30sqrt{2}-60[/latex]
