[tex]Zad. 1\\2x^2-3x+1 < 0\\a > 0 - \text{ramiona paraboli skierowane w gore}\\\Delta=(-3)^2-4*2*1=9-8=1\\\sqrt{\Delta}=1\\x_1=\frac{3-1}{4}=\frac24=\frac12\\x_2=\frac{3+1}4=\frac44=1\\\\f(x) < 0 \text{ dla } x\in(\frac12, 1)[/tex]
[tex]Zad. 2\\P_t=16\sqrt3\\P_t=\frac{a^2\sqrt3}4\\16\sqrt3=\frac{a^2\sqrt3}4 /*4\\64\sqrt3=a^2\sqrt3\\a^2=64\\a=8\\\\r_k=\frac23h_t\\h_t=\frac{a\sqrt3}2\\r_k=\frac23*\frac{a\sqrt3}2=\frac{a\sqrt3}3\\r_k=\frac{8\sqrt3}3\\\\P_k=\pi r^2\\P_k=\pi * (\frac{8\sqrt3}3)^2=\pi*\frac{64*3}9=\pi*\frac{64}3=\frac{64\pi}3[/tex]
[tex]Zad. 3\\a_n=-2n+4\\a_{n+1}=-2(n+1)+4\\a_{n+1}=-2n-2+4\\a_{n+1}=-2n+2\\\\r=a_{n+1}-a_n\\r=-2n+2-(-2n+4)\\r=-2n+2+2n-4\\r=-2\\\\\text{Ciag malejacy}[/tex]
[tex]Zad. 4\\a_2=4\\a_7=20\\\\a_2=a_1+r\\a_7=a_1+6r\\\\\left \{ {{a_1+r=4 /*(-1)} \atop {a_1+6r=20}} \right. \\+\left \{ {{-a_1-r=-4} \atop {a_1+6r=20}} \right. \\-r+6r=-4+20\\5r=16 /:5\\r=\frac{16}5=3\frac15\\\\a_1+3\frac15=4 /-3\frac15\\a_1=\frac45\\\\a_n=\frac45+(n-1)*\frac{16}5\\a_n=\frac45+\frac{16}5n-\frac{16}5\\a_n=\frac{16}5n-\frac{12}5\\a_n=\frac{16n-12}5[/tex]
[tex]Zad. 5\\S_n=a_1*\frac{1-q^n}{1-q}\\a_1=-3\\q=-2\\\\S_8=-3*\frac{1-(-2)^8}{1-(-2)}\\S_8=-3*\frac{1-256}{1+2}\\S_8=-3*\frac{-255}{3}\\S_8=255[/tex]
