[tex]\displaystyle\bf\\2)\\\\Se~da:\\a_1+a_2+a_3+a_4=2020\\r=12\\\\Se~cere:\\\\a_1=?\\\\Rezolvare:\\\\a_1+a_2+a_3+a_4=2020\\a_1+a_1+r+a_1+2r+a_1+3r=2020\\4a_1+6r=2020\\4a_1+6\times12=2020\\4a_1+72=2020\\4a_1=2020-72\\4a_1=1948\\\\a_1=\frac{1948}{4}\\\\\boxed{\bf a_1=487}[/tex]
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5)
[tex]\displaystyle\bf\\Se~da:\\M(6,~2)\\N(-3,~-10)\\\\Se~cere:\\d(M,~N)=?\\\\Rezolvare:\\\\d(M,~N)=\sqrt{(6-(-3))^2+(2-(-10))^2}=\\\\=\sqrt{(6+3)^2+(2+10)^2}=\sqrt{(9)^2+(12)^2}=\\\\=\sqrt{81+144}=\sqrt{225}=15\\\\\boxed{\bf d(M,~N)=15}[/tex]
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6)
[tex]\displaystyle\bf\\Se~da:\\\Delta ABC~cu:\\AB=2\\m(\sphericalangle C)=45^o\\\\Se~cere:\\R=?\\\\Rezolvare:\\Folosim~Teorema~sinusurilor:\\\\\frac{AB}{sin C}=2R\\\\2R=\frac{AB}{sin C}\\\\2R=\frac{2}{sin 45^o}\\\\2R=\frac{~~~2~~~}{\dfrac{\sqrt{2}}{2}}\\\\2R=2\times\dfrac{2}{\sqrt{2}}\\\\2R=\frac{4}{\sqrt{2}}~~~\Big|:2\\\\R=\frac{2}{\sqrt{2}}\\\\R=\frac{2\sqrt{2}}{2}\\\\\boxed{\bf R=\sqrt{2}}[/tex]
