[tex]Zad. 1\\\\a_5=54\\q=3\\\\a_1*q^4=a_5\\a_1*3^4=54\\a_1*81=54 /:81\\a_1=\frac{2}{3}\\\\\underline{Odp. A}[/tex]
[tex]Zad. 2\\\\\sqrt[2]{\frac29*\frac98}=\sqrt{\frac28}=\frac{\sqrt2}{2\sqrt2}=\frac{2}{4}=\frac12\\\\\underline{Odp. C}[/tex]
[tex]Zad. 3\\\\a_n=54*(-\frac13)^n\\a_1=54*(-\frac13)^1=54*(-\frac13)=-\frac{54}3=-18\\a_2=54*(-\frac13)^2=54*\frac19=\frac{54}9=6\\\\a_1*q=a_2\\-18*q=6 /:(-18)\\q=-\frac13\\\\S_4=a_1*\frac{1-q^4}{1-q}\\S_4=-18*\frac{1-(-\frac13)^4}{1-(-\frac13)}\\S_4=-18*\frac{1-\frac1{81}}{\frac43}\\S_4=-18*\frac{\frac{80}{81}}{\frac43}\\S_4=-18*\frac{80}{81}*\frac34\\S_4=-2*\frac{20}{3}*\frac11\\S_4=-\frac{40}3\\S_4=-13\frac13\\\\S_4\in < -14; -12 > \\\\\underline{Odp. C}[/tex]
[tex]Zad. 4\\\\a_3=-16\\a_4=4\\a_3*q=a_4\\-16*q=4 /:(-16)\\q=-\frac14\\\\a_5=a_4*q\\\underline{a_5=4*(-\frac14)=-1}\\a_6=a_5*q\\\underline{a_6=-1*(-\frac14)=\frac14}\\a_7=a_6*q\\\underline{a_7=\frac14*(-\frac14)=-\frac1{16}}\\a_8=a_7*q\\\underline{a_8=-\frac1{16}*(-\frac14)=\frac1{64}}[/tex]
[tex]Zad. 5\\\\a_6=32\\a_{10}=2\\a_{10}=a_6*q^{10-6}\\2=32*q^4 /:32\\\frac1{16}=q^4\\q_1=\frac12 \text{ v } q_2=-\frac12\\a_1=1024 \text{ v } a_1'=-1024\\\\[/tex]
[tex]\text{Istnieja 2 rozwiazania: }\\\text{Dla pierwszego wyrazu rownego } 1024 \text{ i ilorazu rownego } \frac12\text{, ciag jest malejacy}\\\text{Dla pierwszego wyrazu rownego } -1024 \text{ i ilorazu rownego } -\frac12\\\text{ciag nie jest ani rosnacy, ani malejacy, ani staly}[/tex]
[tex]Zad. 6\\\\a)\\a_1=-20\\q=-\frac12\\\\S_5=a_1*\frac{1-q^5}{1-q}\\S_5=-20*\frac{1-(-\frac12)^5}{1-(-\frac12)}\\S_5=-20*\frac{1+\frac1{32}}{1+\frac12}\\S_5=-20*\frac{\frac{33}{32}}{\frac32}\\S_5=-20*\frac{33}{32}*\frac23\\S_5=-5*\frac{11}{4}*\frac11\\S_5=-\frac{55}4\\S_5=-13\frac34[/tex]
[tex]b)\\\\a_2=-\frac1{16}\\a_3=\frac18\\\\a_2*q=a_3\\-\frac1{16}*q=\frac18 /*(-16)\\q=-2\\\\a_1*q=a_2\\a_1*(-2)=-\frac1{16} /*(-\frac12)\\a_1=\frac1{32}\\\\S_7=a_1*\frac{1-q^7}{1-q}\\S_7=\frac1{32}*\frac{1-(-2)^7}{1-(-2)}\\S_7=\frac1{32}*\frac{1+128}{1+2}\\S_7=\frac1{32}*\frac{129}3\\S_7=\frac1{32}*43\\S_7=\frac{43}{32}=1\frac{11}{32}[/tex]
[tex]Zad. 7\\\\a_1=6\\q=-2\\\\S_n=258\\\\258=6*\frac{1-(-2)^n}{1-(-2)} /:6\\43=\frac{1-(-2)^n}{3} /*3\\129=1-(-2)^n /-1\\128=-(-2)^n /*(-1)\\-128=(-2)^n\\n=7\\\\\underline{\text{Nalezy dodac do siebie 7 poczatkowych wyrazow tego ciagu}}[/tex]
