Explicație pas cu pas:
[tex]E(x) = (2x-1)^{2} - (2 - x)(2x + 5) - (x + 3)^{2} + x(9 - 4x) + 18 \\ [/tex]
[tex]= 4 {x}^{2} - 4x + 1 - 4x - 10 + 2{x}^{2} + 5x - {x}^{2} - 6x - 9 + 9x - 4 {x}^{2} +18 \\ [/tex]
[tex]= {x}^{2}[/tex]
[tex]S = E( {2}^{0} ) + E( {2}^{1} ) + E( {2}^{2} ) + E( {2}^{3} ) + … + E( {2}^{50} ) \\[/tex]
[tex]= ({2}^{0})^{2} + {({2}^{1})}^{2} + {({2}^{2})}^{2} + {({2}^{3})}^{2} + ... +{({2}^{50})}^{2} \\ [/tex]
[tex]= {2}^{0} + {2}^{2} + {2}^{4} + {2}^{6} + ... + {2}^{100} \\[/tex]
[tex]= \frac{ {2}^{102} - 1}{3} \\[/tex]
cunoaștem formula:
[tex]{2}^{n} = {2}^{n + 1} - 1[/tex]
⇒
[tex]S_{100} = {2}^{0} + {2}^{1} + {2}^{2} + {2}^{3} + ... + {2}^{100} = {2}^{101} - 1\\[/tex]
[tex]S = S_{100} - ({2}^{1} + {2}^{3} + {2}^{5} + ... + {2}^{99} + {2}^{101} ) + {2}^{101} \\ S = S_{100} - 2({2}^{0} + {2}^{2} + {2}^{4} + {2}^{6} + ... + {2}^{100}) + {2}^{101} \\ [/tex]
[tex]S = S_{100} - 2S + {2}^{101} \\ 3S = S_{100} + {2}^{101} < = > 3S = {2}^{101} - 1 + {2}^{101} \\ 3S = 2 \times {2}^{101} - 1 < = > 3S = {2}^{102} - 1 \\ = > S = \frac{{2}^{102} - 1}{3} [/tex]
