Tổng và tích (2)


Bài 1. Tính tổng \dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\cdots+\dfrac{1}{99.100}.
Bài 2. Tính tổng S=1+2+3+4+\cdots+50.

Bài 3. Tính tích \left(1-\dfrac{1}{2^2}\right).\left(1-\dfrac{1}{3^2}\right).\left(1-\dfrac{1}{4^2}\right).\cdots.\left(1-\dfrac{1}{100^2}\right).

Bài 4. Tính A=\dfrac{1}{\sqrt{1}+\sqrt{2}}+\dfrac{1}{\sqrt{2}+\sqrt{3}}+\cdots+\dfrac{1}{\sqrt{17}+\sqrt{18}}.
Bài 5. Tính tổng
\dfrac{1}{1\sqrt{2}+2\sqrt{1}}+\dfrac{1}{2\sqrt{3}+3\sqrt{2}}+\dfrac{1}{3\sqrt{4}+4\sqrt{3}}+\cdots+\dfrac{1}{17\sqrt{18}+18\sqrt{17}}.
Bài 6. Tính tích
\dfrac{2^3-1}{2^3+1}.\dfrac{3^3-1}{3^3+1}.\dfrac{4^3-1}{4^3+1}.\cdots.\dfrac{22^3-1}{22^3+1}.
Bài 7. Tính tổng
1!(1^2+1+1)+2!(2^2+2+1)+3!(3^2+3+1)+\cdots+100!(100^2+100+1).
Bài 8. Tính tích \left(1+\dfrac{1}{2^{2^0}}\right).\left(1+\dfrac{1}{2^{2^1}}\right).\left(1+\dfrac{1}{2^{2^2}}\right).\cdots.\left(1+\dfrac{1}{2^{2^{11}}}\right).
Bài 9. Tính tổng
\sqrt{1+\dfrac{1}{1^2}+\dfrac{1}{2^2}}+\sqrt{1+\dfrac{1}{2^2}+\dfrac{1}{3^2}}+\sqrt{1+\dfrac{1}{3^2}+\dfrac{1}{4^2}}+\cdots+\sqrt{1+\dfrac{1}{11^2}+\dfrac{1}{12^2}}.
Bài 10. Tính tổng
\dfrac{x_0^3}{1-3x_0+3x_0^2}+\dfrac{x_1^3}{1-3x_1+3x_1^2}+\dfrac{x_2^3}{1-3x_2+3x_2^2}+\cdots+\dfrac{x_{101}^3}{1-3x_{101}+3x_{101}^2}.
Ở đây x_i=\dfrac{i}{101}\,\,\,\,\forall i=0,1,2,\cdots,101.

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