Zadania 1,3,4,7 z kartki reszta nie ważna. Na dzisiaj mi to potrzebne.  Z góry dzięki za pomoc ;)

Mam nadzieje, że da się rozczytać :D 

[latex]1.\Ciag a_n jest geometryczny jezeli iloraz a_{n+1}:a_n jest staly.\\a) a_n=frac{5}{2^n}; a_{n+1}=frac{5}{2^{n+1}}\\a_{n+1}:a_n=frac{5}{2^{n+1}}:frac{5}{2^n}=frac{5}{2^{n+1}}cdotfrac{2^n}{5}=2^{n-(n+1)}\\=2^{n-n-1}=2^{-1}=frac{1}{2}=const.\\b) b_n=-frac{1}{3}cdot2^n; b_{n+1}=-frac{1}{3}cdot2^{n+1}\\b_{n+1}:b_n=-frac{1}{3}cdot2^{n+1}:(-frac{1}{3}cdot2^n)=2^{n+1-n}=2^1=2=const.[/latex] [latex]c) c_n=4^{2n+1}; c_{n+1}=4^{2(n+1)+1}=4^{2n+2+1}=4^{2n+3}\\c_{n+1}:c_n=4^{2n+3}:4^{2n+1}=4^{2n+3-(2n+1)}=4^{2n+3-2n-1}\\=4^2=16=const.[/latex] [latex]2.\a; b; c - ciag geometryczny, wtedy acdot c=b^2\\a) 2; x; 32; y\\x^2=2cdot32 o x^2=64 o x=pmsqrt{64} o x=8 vee x=-8\\xcdot y=32^2\\pm8y=1024 /:(pm8) o y=pm128[/latex] [latex]b) x; frac{1}{10}; y; 10\\y^2=frac{1}{10}cdot10 o y^2=1 o y=pmsqrt1 o y=1 vee y=-1\\xcdot y=(frac{1}{10})^2\\pm1x=frac{1}{100} o x=pmfrac{1}{100}[/latex] [latex]4.\1; a; b; c; frac{81}{256}\\1-pierwszy wyraz ciagu\\frac{81}{256}-piaty wyraz ciagu\\a_5:a_1=q^4\\q^4=frac{81}{256}:1\\q=pmsqrt[4]{frac{81}{256}}\\q=pmfrac{3}{4}\\a=1cdot(pmfrac{3}{4})=pmfrac{3}{4}\\b=aq o b=pmfrac{3}{4}cdot(pmfrac{3}{4})=frac{9}{16}\\c=bq o c=frac{9}{16}cdot(pmfrac{3}{4})=pmfrac{27}{64}[/latex] [latex]7.\frac{1}{8}+frac{3}{8}+frac{9}{8}+...+frac{729}{8}\\a_1=frac{1}{8}; a_2=frac{3}{8}\\q=a_2:a_! o q=frac{3}{8}:frac{1}{8}=frac{3}{8}cdotfrac{8}{1}=3\\a_n=a_1q^{n-1} ofrac{1}{8}cdot3^{n-1}=frac{729}{8} /cdot8\\3^{n-1}=729\\3^{n-1}=3^6iff n-1=6 o n=6+1 o n=7\\S_7=frac{a_1(1-q^n)}{1-q} o S_7=frac{frac{1}{8}cdot(1-3^7)}{1-3}=frac{1}{8}cdotfrac{1-2187}{-2}=frac{-2186}{-16}=frac{1093}{8}=136frac{5}{8}[/latex]