3.
[tex]n=20\\a_1=1,5\\r=-2.4\\a_{20}=a_1+19r\\a_{20}=1,5+19*(-2,4)=1,5-45.6=-44.1\\S_{20}=\frac{a_1+a_{20}}2*20=10(a_1+a_{20})\\S_{20}=10(1,5-44,1)=10*-426[/tex]
4.
[tex]a_1=-4\\a_2=2\\a_3=-1\\a_4=a\\a_2=a_1*q\\2=(-4)*q /:(-4)\\-\frac24=q\\q=-\frac12\\\\a_4=a_1*q^3\\a=(-4)*(-\frac12)^3=(-4)*(-\frac18)=\frac12[/tex]
5.
[tex]a_5=a_6*q^{-1}\\a_7=a_6*q\\a_5*a_7=169\\a_6*q^{-1}*a_6*q=169\\a_6^2*q^{-1+1}=169\\a_6^2*q^0=169\\a_6^2*1=169\\a_6=13[/tex]
6.
[tex]q > 0\\a_1=10\\a_2=x\\a_3=30\\a_3=a_1*q^2\\30=10*q^2 /:10\\3=q^2\\q=\sqrt3\\x=a_1*q\\x=10*\sqrt3=10\sqrt3\\\\S_8=a_1*\frac{1-q^8}{1-q}\\S_8=10*\frac{1-(\sqrt3)^8}{1-\sqrt3}\\S_8=10*\frac{-80}{1-\sqrt3}\\S_8=\frac{-800}{1-\sqrt3}*\frac{1+\sqrt3}{1+\sqrt3}=\frac{-800(1+\sqrt3)}{1-3}=\frac{-800(1+\sqrt3)}{-2}=400(1+\sqrt3)=400+400\sqrt3[/tex]
7.
[tex]\text{Od najwiekszej do najmniejszej - ciag malejacy: } 0 < q < 1[/tex]
[tex]a_1, a_2, a_3 - \text{ ciag geometryczny}\\a_2=a_1q\\a_3=a_1q^2\\\\a_1+a_2+a_3=78\\a_1+a_1q+a_1q^2=78\\a_1(1+q+q^2)=78 /:(1+q+q^2)\\a_1=\frac{78}{1+q+q^2}[/tex]
[tex]b_1, b_2, b_3 - \text{ ciag arytmetyczny}\\b_1=a_1\\b_2=a_2+12=a_1q+12\\b_3=a_3=a_1q^2\\\\b_2=\frac{b_1+b_3}2 /*2\\2b_2=b_1+b_3\\2(a_1q+12)=a_1+a_1q^2\\2a_1q+24=a_1+a_1q^2\\24=a_1-2a_1q+a_1q^2\\24=a_1(1-2q+q^2)[/tex]
[tex]24=\frac{78}{1+q+q^2}*(1-2q+q^2)\\24=\frac{78(1-2q+q^2)}{1+q+q^2} /*1+q+q^2\\24(1+q+q^2)=78(1-2q+q^2)\\24+24q+24q^2=78-156q+78q^2\\24q^2-78q^2+24q+156q+24-78=0\\-54q^2+180q-54=0\\\Delta=32400-4*(-54)*(-54)=32400-11664=20736\\\sqrt{\Delta}=144\\q_1=\frac{-180-144}{-108}=\frac{324}{108}=3 - \text{ nie zgadza sie z zalozeniem } 0 < q < 1 \text{ - ciag bylby rosnacy}\\q_2=\frac{-180+144}{-108}=\frac{-36}{-108}=\frac13[/tex]
[tex]a_1=\frac{78}{1+\frac13+(\frac13)^2}=\frac{78}{\frac43+\frac19}=\frac{78}{\frac{13}9}=78*\frac9{13}=6*9=54\\a_2=54*\frac13=18\\a_3=6[/tex]
Odp. Te liczby to 54, 18, 6
