Răspuns:
[tex](a)\boldsymbol{ \red{4048}}, (b)\boldsymbol{ \red{1}}[/tex]
Explicație pas cu pas:
Scriem expresia într-o formă mai simplă:
[tex]E(x) = (2x -3)^2 - 3(x - 1)(x + 1) - 5(2 - 3x) = 4x^2 - 12x + 9 - 3(x^2 - 1) - 10 + 15x = 4x^2 - 12x + 9 - 3x^2 + 3 - 10 + 15x = x^2 + 3x + 2 = (x + 1)(x + 2)[/tex]
[tex]a) \ E(2023) - E(2022) = (2023 + 1)(2023 + 2) - (2022 + 1)(2022 + 2) = 2024 \cdot 2025 - 2023 \cdot 2024 = 2024(2025 - 2023) = 2024 \cdot 2 = \bf4048[/tex]
b) Folosim formula:
[tex]\boxed{\boldsymbol{\dfrac{1}{n \cdot (n + 1)} = \dfrac{1}{n} - \dfrac{1}{n + 1}}}[/tex]
[tex]\dfrac{1}{E(0)} + \dfrac{1}{E(1)} + ... + \dfrac{1}{E(8)} + 0,1 = \\[/tex]
[tex]= \dfrac{1}{(0+1)(0+2)} + \dfrac{1}{(1+1)(1+2)} + ... + \dfrac{1}{(8+1)(8+2)} + 0,1 \\[/tex]
[tex]= \dfrac{1}{1 \cdot 2} + \dfrac{1}{2 \cdot 3} + ... + \dfrac{1}{9 \cdot 10} + 0,1 \\[/tex]
[tex]= \dfrac{1}{1} - \dfrac{1}{2} + \dfrac{1}{2} - \dfrac{1}{3} + ... + \dfrac{1}{9} - \dfrac{1}{10} + 0,1 \\[/tex]
Se reduc termenii asemenea
[tex]= \dfrac{1}{1} - \dfrac{1}{10} + 0,1 = 1 - 0,1 + 0,1 = \bf 1 \in \Bbb{N}\\[/tex]
