[latex]4.\a) a_n=frac{1}{2}n^2-18\\frac{1}{2}n^2-18=0\\frac{1}{2}n^2=18 /cdot2\\n^2=36\\n=sqrt{36}\\n=6\\a_6=0[/latex] [latex]b) a_n=frac{n^2-4n}{n+1}\\frac{n^2-4n}{n+1}=0iff n^2-4n=0\\n(n-4)=0\\n=0 vee n=4\\a_4=0[/latex] [latex]c) a_n=n^2-6n+8\\n^2-6n+8=0\\Delta=(-6)^2-4cdot1cdot8=36-32=4; sqrtDelta=sqrt4=2\\n_1=frac{6-2}{2cdot1}=2 vee frac{6+2}{2cdot1}=4\\a_2=a_4=0[/latex] [latex]d) a_n=frac{n^2-4n+3}{n+1}\\frac{n^2-4n+3}{n+1}=0iff n^2-4n+3=0\\Delta=(-4)^2-4cdot1cdot3=16-12=4; sqrtDelta=sqrt4=2\\n_1=frac{4-2}{2cdot1}=1 vee n_2=frac{4+2}{2cdot1}=3\\a_1=a_3=0[/latex] [latex]e) a_n=(n^2-49)(n^2-6n)\\(n^2-49)(n^2-6n)=0\\n^2-49=0 vee n^2-6n=0\\n^2=49 vee n(n-6)=0\\n=7 vee n=0 vee n=6\\a_6=a_7=0[/latex] [latex]f) a_n=frac{n^3-4n^2+4n}{n^2+4}\\frac{n^3-4n^2+4n}{n^2+4}=0iff n^3-4n^2+4n=0\\n(n^2-4n+4)=0\\n(n-2)^2=0\\n=0 vee n=2\\a_2=0[/latex] [latex]5.\a) a_n=-2n+9\\-2n+9 > 0\\-2n > -9 /:(-2)\\n < 4,5\\Dodatnie:a_1; a_2; a_3; a_4[/latex] [latex]b) a_n=10-n^2\\10-n^2 > 0\\-n^2 > -10 /cdot(-1)\\n^2 < 10\\n < sqrt{10} wedge n > -sqrt{10}\\3 < sqrt{10} < 4\\Wieksze od zera:a_1; a_2; a_3[/latex] [latex]c) a_n=n^2+3n-10\\n^2+3n-10 > 0\\Delta=3^2-4cdot1cdot(-10)=9+40=49; sqrtDelta=sqrt{49}=7\\n_1=frac{-3-7}{2cdot1}=-5; n_2=frac{-3+7}{2cdot1}=2\\Ramiona paraboli skierowane w gore oraz ninmathbb{N^+}\\Wyrazy dodatnie:a_3; a_4; a_5;...[/latex] [latex]6.\a) a_n=frac{3}{2}n-9\\frac{3}{2}n-9 < 0\\frac{3}{2}n < 9 /cdotfrac{2}{3}\\n < 6\\Wyrazy ujemne:a_1; a_2; a_3; a_4; a_5[/latex] [latex]b) a_n=-2n+5\\-2n+5 < 0\\-2n < -5 /:(-2)\\n > 2,5\\Wyrazy ujemne:a_3; a_4; a_5; a_6;...[/latex] [latex]c) a_n=n^2-12\\n^2-12 < 0\\n^2 < 12\\n < sqrt{12} wedge n > -sqrt{12} wedge ninmathnn{N^+}\\3 < sqrt{12} < 4\\Wyrazy ujemne:a_1; a_2; a_3[/latex] [latex]d) a_n=n^2-5n\\n^2-5n < 0\\n(n-5) < 0\\n=0; n=5\\ramiona skierowane w gore i ninmathbb{N^+}\\Wyrazy ujemne:a_1; a_2; a_3; a_4[/latex] [latex]e) a_n=n^2-11n+10\\n^2-11n+10 < 0\\Delta=(-11)^2-4cdot1cdot10=121-40=81; sqrtDelta=sqrt{81}=9\\n_1=frac{11-9}{2cdot1}=1; n_2=frac{11+9}{2cdot1}=10\\ramiona skierowane w gore oraz ninmathbb{N^+}\\Wyrazy ujemne:a_2; a_3; a_4; a_5; a_6; a_7; a_8; a_9[/latex] [latex]f) a_n=3n^2-10n+8\\3n^2-10n+8 < 0\\Delta=(-10)^2-4cdot3cdot8=100-96=4; sqrtDelta=sqrt4=2\\n_1=frac{10-2}{2cdot3}=frac{4}{3}; n_2=frac{10+2}{2cdot3}=2\\ramiona skierowane w gore oraz ninmathbb{N^+}\\Wyrazy ujemnych brak.[/latex]
