a,a+r,a+2r - kolejne wyrazy ciągu arytmetycznego a+1,a+r+3,a+2r+7 - kolejne wyrazy ciągu geometrycznego Wyznaczam [latex]a[/latex] [latex]a+a+r+a+2r=27[/latex] [latex]3a+3r=27 /:3[/latex] [latex]a+r=9[/latex] [latex]a=9-r[/latex] Obliczam [latex]a[/latex] [latex](a+1)(a+2r+7)=(a+r+3)^2[/latex] [latex](9-r+1)(9-r+2r+7)=(9-r+r+3)^2[/latex] [latex](10-r)(16+r)=12^2[/latex] [latex]160+10r-16r-r^2=144[/latex] [latex]160-6r-r^2=144[/latex] [latex]-r^2-6r+160-144=0[/latex] [latex]-r^2-6r+16=0 /:(-1)[/latex] [latex]r^2+6r-16=0[/latex] [latex]Delta_r=6^2-4 cdot 1 cdot (-16)=36+64=100[/latex] [latex]sqrt{Delta}= sqrt{100}=10[/latex] [latex]r_1=frac{-6-10}{2}=frac{-16}{2}=-8[/latex] [latex]r_2=frac{-6+10}{2}=frac{4}{2}=2[/latex] Obliczam [latex]a[/latex] [latex]a=9-r[/latex] [latex]a=9-(-8)=9+8=17[/latex] lub [latex]a=9-r=9-2=7[/latex] Wyrazy ciągu: [latex]a_1=17[/latex] [latex]a_2=17-8=9[/latex] [latex]a_3=17+2 cdot (-8)=17-16=1[/latex] lub [latex]a_1=7[/latex] [latex]a_2=7+2=9[/latex] [latex]a_3=7+2 cdot 2=7+4=11[/latex]
