[latex]Zad. 7.106 a)\1+2+4+...+64\a_n=a_1cdot q^{n-1}\a_1=1\q=4:2=2\a_4=1cdot 2^3=8\a_5=16\a_6=32\1+2+4+8+16+32+64=127\\S_n=a_1cdot frac{1-q^n}{1-q}\S_7=1cdot frac{1-2^7}{1-2}=1cdot frac{-127}{-1})=127\---------------------------[/latex] [latex]Zad 7.107\ a_3=45\a_6=1215\S_8=?\\a_3=a_1cdot q^2\a_6=a_1cdot q^5\\45=a_1cdot q^2\1215=a_1cdot q^5\\a_1=frac{45}{q^2}\1215=frac{45}{q^2}cdot q^5 ||frac{q^5}{q^2}=q^3[/latex] [latex]1215=45cdot q^3\27=q^3\q=3\a_1= frac{45}{9}=5[/latex] [latex]S_8=5cdot frac{1-3^8}{1-3}=5cdot frac{-6560}{-2}=5cdot 3280=16400[/latex]
a) [latex]a_1=1 \ q=frac{a_2}{a_1}=frac21=2 \ a_n=64 [/latex] tyle wiemy. Nie wiemy ile wyrazów tego ciągu zsumowano. Nie znamy [latex]n[/latex]. Aby skorzystać ze wzoru na sumę ciągu geom, trzeba to [latex]n[/latex] obliczyć. Można to policzyć wykorzystując wzor na n-ty wyraz ciągu geom: [latex]a_n=a_1cdot q^{n-1}[/latex] zatem [latex]64=1cdot 2^{n-1} \ 2^{n-1}=64 \ 2^{n-1}=2^6 \ n-1=6 \ n=7[/latex] a więc [latex]S_n=a_1cdot frac{1-q^n}{1-q} \ S_7=1cdot frac{1-2^7}{1-2}=frac{1-128}{-1}=frac{-127}{-1}=127[/latex] b) [latex]a_1=1 \ q=frac{a_2}{a_1}=frac{-3}1=-3 \ a_n=729 \ a_n=a_1cdot q^{n-1} \ 729=(-3)^{n-1} \ 3^6=(-3)^{n-1} \ 3^6=3^{n-1} \ 6=n-1 \ n=7 \ S_7=1cdot frac{1-(-3)^7}{1-(-3)}= \ =frac{1-(-2187)}4=547[/latex] c) [latex]n=8 \ a_1=32 \ a_8=-0.25 \ q=frac{-16}{32}=-frac12 \ S_8=32cdot frac{1-(-frac12)^8}{1-(-frac12)}=32cdot frac{1-frac1{256}}{frac32}=32cdot frac{255}{256}cdot frac23=32cdotfrac{85}{128}=21frac14[/latex] d) [latex]a_1=1frac12=frac32 \ q=frac1{1frac12} = frac1{frac32}=frac23 \ a_n=frac{32}{243} \ a_n=a_1cdot q^{n-1} \ frac{32}{243}=frac32cdot (frac23)^{n-1} \ frac{32}{243}=(frac23)^{-1}cdot (frac23)^{n-1} \ (frac23)^5=(frac23)^{n-2} \ 5=n-2 \ n=7 [/latex] [latex]S_7=frac32cdot frac{1-(frac23)^7}{1-frac23}=frac32cdot frac{1-frac{128}{2187}}{frac13}=frac32cdot frac{frac{2059}{2187}}{frac13}=frac32cdot 3cdot frac{2059}{2187}=frac{2059}{486}[/latex] 7.107 [latex]a_3=45, a_6=1215 \ S_8=? \ \ a_6=a_3cdot q^3 \ 1215=45cdot q^3 \ q^3=frac{1215}{45} \ q^3=27 o q=3 \ a_3=a_1cdot q^2 \ a_1=frac{a_3}{q^2} \ a_1=frac{45}9=5 \ S_8=5cdot frac{1-3^8}{1-3}=5cdot{1-6561}{-2} = 5cdot frac{-6560}{-2}=5cdot 3280=16400[/latex]
