[latex]frac{1}{ 4^{x} + 11 }[/latex] [latex]2^{x} - 1[/latex] [latex]16^{x} - 13[/latex] a) Wyznacz x [latex]frac{1}{4^x+11} cdot left(16^x-13 ight)=(2^x-1)^2[/latex] [latex]frac{16^x-13}{4^x+11} =(2^x-1)^2[/latex] [latex]frac{(2^4)^x-13}{(2^2)^x+11} = (2^x-1)^2[/latex] [latex]frac{(2^x)^4-13}{(2^x)^2+11} = (2^x-1)^2[/latex] podstawiamy: [latex]2^x=t,t>0[/latex] [latex]frac{t^4-13}{t^2+11} = (t-1)^2[/latex] [latex]frac{t^4-13}{t^2+11} = t^2-2t+1[/latex] [latex]t^4-13=(t^2-2t+1)(t^2+11)[/latex] [latex]t^4-13=t^4+11t^2-2t^3-22t+t^2+11[/latex] [latex]t^4-13-t^4-11t^2+2t^3+22t-t^2-11=0[/latex] [latex]2t^3 - 12t^2 + 22t - 24=0 /:2[/latex] [latex]t^3 - 6t^2 + 11t - 12=0[/latex] [latex]t^3 - 4t^2 -2t^2+ 8t+3t - 12=0[/latex] [latex]t^2(t-4) -2t(t-4)+3(t-4)=0[/latex] [latex](t-4)(t^2 -2t+3)=0[/latex] [latex]Delta=(-2)^2-4 cdot 1 cdot 3=4-12=-8<0[/latex] [latex]t-4=0[/latex] [latex]t=4[/latex] [latex]2^x=4[/latex] [latex]2^x=2^2[/latex] [latex]x=2[/latex] ================= b) Napisz wyraz ogólny ciągu. [latex]a_1=frac{1}{ 4^{x} + 11 }=frac{1}{ 4^2+ 11 }=frac{1}{ 16+ 11 }=frac{1}{27}[/latex] [latex]a_2=2^{x} - 1=2^2 - 1=4-1=3[/latex] [latex]q= frac{a_2}{a_1} = frac{3}{frac{1}{27}}=3 cdot 27=81[/latex] [latex]a_n= frac{1}{27} cdot 81^{n-1}= frac{1}{3^3} cdot (3^4)^{n-1}=[/latex] [latex]3^{-3} cdot 3^{4n-4}=3^{4n-7}[/latex]
