Zad.1 a) [latex]2^-^3=(frac{1}{2})^3=frac{1}{8}[/latex] b) [latex]0,5^-^4=(frac{1}{2})^-^4=2^4=16[/latex] c) [latex](1frac{4}{9})^-^2=(frac{13}{9})^-^2=(frac{9}{13})^2=frac{81}{69}[/latex] d) [latex]frac{7^-^7:7^-^1^1}{7^3*7^-^3}=frac{7^-^3}{7^0}=7^-^3=(frac{1}{7})^3=frac{1}{343}[/latex] e) [latex]27^frac{1}{3} =sqrt[3]{27}=3[/latex] f) [latex]4^frac{3}{2}=sqrt{4^3}=sqrt{64}=8[/latex] g) [latex]0,001^-^frac{4}{3}=sqrt[3]{0,001^-^4}=sqrt[3]{1000^4}=10000[/latex] h) [latex](2frac{1}{4})^frac{3}{2}=(frac{9}{4})^frac{3}{2}=sqrt{(frac{9}{4})^3}=sqrt{frac{729}{64}}=frac{27}{8}[/latex] Zad.2 a) [latex]log_216=log_22^4=4[/latex] b) [latex]log_3frac{1}{9}=log_39^-^1=log_3(3^2)^-^1=log_33^-^2=-2[/latex] c) [latex]log125+log4-log5=log(125*4:5)=log100[/latex] d) [latex]log_3(log_5125)=log_3(log_55^3)=log_33=1[/latex] e) [latex]log_525-log_216=log_55^2-log_22^4=2-4=-2[/latex] f) [latex]log_2(2sqrt{2})=log_22+log_2sqrt{2}=1+log_22^frac{1}{2}=1+frac{1}{2}=1frac{1}{2}[/latex] Zad.3 [latex]a_n=frac{4n-1}{n^2}[/latex] [latex]a_1=frac{4*1-1}{1^2}=frac{4-1}{1}=3[/latex] [latex]a_2=frac{4*2-1}{2^2}=frac{8-1}{4}=frac{7}{4}=1frac{3}{4}[/latex] [latex]a_3=frac{4*3-1}{3^2}=frac{12-1}{9}=frac{11}{9}=1frac{2}{9}[/latex] [latex]a_1_0=frac{4*10-1}{10^2}=frac{40-1}{100}=frac{39}{100}[/latex] [latex]a_5_0=frac{4*50-1}{50^2}=frac{200-1}{2500}=frac{199}{2500}[/latex] Zad.4 Dane: Oblicz: [latex]a_2=13[/latex] [latex]a_1=?[/latex] [latex]a_7=37[/latex] [latex]r=?[/latex] [latex]S_2_0=?[/latex] [latex]a_n=a_1+(n-1)*r[/latex] [latex]a_3=a_1+(3-1)*r[/latex] [latex]13=a_1+2r[/latex] [latex]a_7=a_1+(7-1)*r[/latex] [latex]37=a_1+6r[/latex] [latex] left { {{a_1+2r=13} atop {a_1+6r=37}} ight.[/latex] [latex] left { {{a_1=-2r+13} atop {a_1+6r=37}} ight.[/latex] [latex]a_1+6r=37[/latex] [latex]-2r+13+6r=37[/latex] [latex]4r=37-13[/latex] [latex]4r=24/:4[/latex] [latex]r=6[/latex] [latex]a_1=-2r+13=-2*6+13=-12+13=1[/latex] [latex]a_n=a_1+(n-1)*r[/latex] [latex]a_2_0=1+(20-1)*6=1+19*6=1+114=115[/latex] [latex]S_n=frac{a_1+a_n}{2}*n[/latex] [latex]S_2_0=frac{a_1+a_2_0}{2}*20=frac{1+115}{2}*20=frac{116}{2}*20=58*20=1160[/latex] Zad.5 -3,9,-27,81,... [latex]a_1=-3[/latex] [latex]q=frac{a_2}{a_1}=frac{9}{-3}=-3[/latex] a) [latex]a_n=a_1*q^n^-^1[/latex] [latex]a_1_0=a_1*q^1^0^-^1=a_1*q^9=-3*(-3)^9=-3*(-19683)=59049[/latex] b) [latex]S_n=a_1*frac{1-q^n}{1-q}[/latex] [latex]S_1_0=-3*frac{1-(-3)^1^0}{1-(-3)}=-3*frac{1-59049}{1+3}=-3*(-frac{59048}{4})=-3*(-14762)=44286[/latex] Zad.6 Dane: Oblicz: [latex]a_1=30[/latex] [latex]S_1_5=?[/latex] [latex]r=5[/latex] [latex]a_n=a_1+(n-1)*r[/latex] [latex]a_1_5=30+(15-1)*5=30+14*5=30+70=100[/latex] [latex]S_n=frac{a_1+a_n}{2}*n[/latex] [latex]S_1_5=frac{a_1+a_1_5}{2}*15=frac{30+100}{2}*15=frac{130}{2}*15=65*15=975[/latex] Zad.7 [latex]K_0[/latex] - kapitał początkowy [latex]K_n[/latex] - kapitał końcowy r - oprocentowanie roczne n- czas trwania lokaty m- ilość kaitalizacji w roku Dane: Oblicz: [latex]K_0=5000[/latex] [latex]K_n=?[/latex] r=4%=0,04 n=1,5 m=2 [latex]K_n=K_0*(1+frac{r}{m})^n^*^m=5000*(1+frac{0,04}{2})^1^,^5^*^2=5000*(1,02)^3=5000*1,061208=5306,04[/latex]
