[tex]a) \\a_1=\sqrt5-2\\a_2=x\\a_3=\frac{\sqrt5+2}4\\\\a_n=a_1*q^{n-1}\\a_3=a_1*q^2\\\frac{\sqrt5+2}4=(\sqrt5-2)*q^2 /:(\sqrt5-2)\\\frac{\sqrt5+2}4*\frac1{\sqrt5-2}=q^2\\q^2=\frac{\sqrt5+2}{4(\sqrt5-2)}\\q^2=\frac{\sqrt5+2}{4(\sqrt5-2)}*\frac{\sqrt5+2}{\sqrt5+2}\\q^2=\frac{(\sqrt5+2)^2}{4*(5-4)}\\q^2=\frac{(\sqrt5+2)^2}{4*1}\\q^2=\frac{(\sqrt5+2)^2}{2^2}\\q^2=(\frac{\sqrt5+2}2)^2\\q=\frac{\sqrt5+2}2[/tex]
[tex]a_2=a_1*q\\x=(\sqrt5-2)*\frac{\sqrt5+2}2\\x=\frac{(\sqrt5-2)(\sqrt5+2)}2\\x=\frac{5-4}2\\x=\frac12\\\\[/tex]
b)
[tex]q=-\frac23\\a_6=\frac{32}{27}\\\\a_6=a_1*q^{5}\\\frac{32}{27}=a_1*(-\frac23)^5\\\frac{32}{27}=a_1*(-\frac{32}{243}) /*(-\frac{243}{32})\\-\frac{243}{32}*\frac{32}{27}=a_1\\-\frac{243}{27}=a_1\\a_1=-9[/tex]
c)
[tex]a_1=-2\\a_2=4\\a_2=a_1*q\\4=(-2)*q /:(-2)\\\frac4{-2}=q\\q=-2\\\\a_5=a_1=q^4\\a_5=-2*(-2)^4\\a_5=(-2)^5\\a_5=-32\\\\S_n=a_1*\frac{1-q^n}{1-q}\\S_5=-2*\frac{1-(-2)^5}{1-(-2)}\\S_5=-2*\frac{1-(-32)}{1+2}\\S_5=-2*\frac{1+32}3\\S_5=-2*\frac{33}3\\S_5=-2*11\\S_5=-22[/tex]
d)
[tex]S_5=-11\\q=-2\\S_5=a_1*\frac{1-q^5}{1-q}\\-11=a_1*\frac{1-(-2)^5}{1-(-2)}\\-11=a_1*\frac{1-(-32)}{1+2}\\-11=a_1*\frac{1+32}{3}\\-11=a_1*\frac{33}3\\-11=a_1*11 /:11\\-1=a_1\\a_2=a_1*q\\a_2=(-1)*(-2)\\a_2=2[/tex]
e)
[tex]a_n=a_k*q^{n-k}\\a_7=20\\a_9=5\\a_9=a_7*q^{9-7}\\a_9=a_7*q^{2}\\\\5=20*q^2 /:20\\\frac5{20}=q^2\\\frac14=q^2\\q=\frac12 \text{ lub } q=-\frac12\\\\\text{dla }q=-\frac12 \text{, ciag nie jest ani rosnacy ani malejacy ani staly, wiec } q=\frac12[/tex]
