Răspuns
Explicație pas cu pas:
1. [tex]a_{50} = 80 , r= 2\\a_{50} = a_{1} + (n-1) * r \\50 = a_{1} + (50-1) * 2\\a_{1} = 50 - 49*2\\a_{1} = 50 - 98 \\a_{1} = -18\\\\[/tex]
[tex]a_{30} = a_{1} + (30-1)*2\\a_{30} = -18 + 29*2\\a_{30} = -18 + 58\\a_{30} = 40[/tex]
[tex]S_{12} = 12* \frac{2*(-18) + (12-1)*2}{2} = 12* \frac{-36 + 22}{2} = 12* \frac{-14}{2} = -84[/tex]
[tex]S_{100} = 100 * \frac{2*(-18)+(100-1)*2}{2} = 100* \frac{-36+99*2}{2} = 100* \frac{-36+198}{2} = 100* \frac{162}{2} = 8100[/tex]
2. [tex]a_{3} = a_{1} + (n-1) * r \\\\\frac{3}{2} = \frac{-1}{2}+ 2r\\ \\2r = \frac{3}{2} - \frac{-1}{2} \\\\2r = \frac{4}{2} \\\\2r = 2 \\r=1[/tex]
[tex]a_{5} = a_{1} + (n-1) * r\\12 = a_{1} + 4r\\\\a_{19} = a_{1} + (n-1)*r\\19 = a_{1} + 18r[/tex]
[tex]a_{1} = 12 - 4r\\ a_{1} = 19 - 18r\\12-4r = 19-18r\\-4r+18r = 19-12\\-14r = 7\\r = \frac{-7}{14} = \frac{1}{2}[/tex]
[tex]S_{26} = 26 * \frac{2*a_{1}+ (n-1)*r}{2} \\\\1053= 26* \frac{2*33+ (26-1)* r }{2}\\\\1053 = 26* \frac{66+25r}{2} \\\\\frac{66+25r}{2} = \frac{405}{10} \\ \\\frac{66+25r}{2} = \frac{81}{2} \\\\66+25r = 81 \\25r = 15[/tex]
[tex]r = \frac{15}{25} = \frac{3}{5}[/tex]
