Răspuns:
[tex]\boldsymbol{ \red{ a. \ \dfrac{2575}{51}}}[/tex]
Explicație pas cu pas:
[tex]S = \dfrac{3^2 - 1 + 2}{3^2 - 1} + \dfrac{5^2 - 1 + 2}{5^2 - 1} + \dfrac{7^2 - 1 + 2}{7^2 - 1} + ... + \dfrac{101^2 - 1 + 2}{101^2 - 1}\\[/tex]
[tex]S = 1 + \dfrac{2}{(3-1)(3+1)} + 1 + \dfrac{2}{(5-1)(5+1)} + 1 + \dfrac{2}{(7 - 1)(7 + 1)} + ... + 1 + \dfrac{2}{(101-1)(101+1)}\\[/tex]
(101 - 3) : 2 + 1 = 98 : 2 + 1 = 49 + 1 = 50 (termeni)
[tex]S = \underbrace{1 + 1 + 1 + ... + 1}_{50} + \dfrac{2}{2 \cdot 4} + \dfrac{2}{4 \cdot 6} + \dfrac{2}{6 \cdot 8} + ... + \dfrac{2}{100 \cdot 102}\\[/tex]
[tex]S = 50 + \dfrac{4 - 22}{2 \cdot 4} + \dfrac{6 - 4}{4 \cdot 6} + \dfrac{8 - 6}{6 \cdot 8} + ... + \dfrac{102 - 100}{100 \cdot 102}\\[/tex]
[tex]S = 50 + \dfrac{1}{2} - \dfrac{1}{4} + \dfrac{1}{4} - \dfrac{1}{6} + \dfrac{1}{6} - \dfrac{1}{8} + ... + \dfrac{1}{100} - \dfrac{1}{102} \\[/tex]
[tex]S = 50 + \dfrac{1}{2} - \dfrac{1}{102} = \dfrac{50 \cdot 102 + 51 - 1}{102} = \dfrac{5150}{102}^{(2}\\[/tex]
[tex]\bf S = \dfrac{2575}{51}[/tex]
