Tính: 1xx+1+1x+1x+2+1x+2x+3+1x+3x+4+1x+4x+5+1x+5

1xx+1+1x+1x+2+.....+1x+4x+5−1x+5=1x−1x+1+......+1x+4−1x+5+1x+5=1x

Tính: 1/x(x+1) + 1/(x+1)(x+2) + 1(x+2)(x+3) + 1/(x+3)(x+4)

Câu 1

Cộng các phân thức sau: $\frac{1}{x (x - y) (x - z)} + \frac{1}{y (y - z) (y - x)} + \frac{1}{z (z - x) (z - y)}$

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Lời giải

$\begin{array}{l} \frac{1}{x (x - y) (x - z)} + \frac{1}{y (y - z) (y - x)} + \frac{1}{z (z - x) (z - y)} \\ = \frac{1}{x (x - y) (x - z)} - \frac{1}{y (y - z) (x - y)} + \frac{1}{z (x - z) (y - z)} \\ = \frac{yz (y - z) - xz (x - z) + xy (x - y)}{xyz (x - y) (y - z) (x - z)} \\ = \frac{y^{2} z - {yz}^{2} - x^{2} z + {xz}^{2} + x^{2} y - {xy}^{2}}{xyz (x - y) (y - z) (x - z)} = \frac{- z (x^{2} - y^{2}) + z^{2} (x - y) + xy (x - y)}{xyz (x - y) (y - z) (z - x)} \\ = \frac{(x - y) [- z (x + y) + z^{2} + xy]}{xyz (x - y) (y - z) (x - z)} = \frac{(x - y) (- xz - yz + z^{2} + xy)}{xyz (x - y) (y - z) (x - z)} \\ = \frac{(x - y) [- z (x - z) + y (x - z)]}{xyz (x - y) (y - z) (x - z)} = \frac{(x - y) (y - z) (x - z)}{xyz (x - y) (y - z) (x - z)} = \frac{1}{xyz} \end{array}$

Câu 2

Cộng các phân thức sau: $\frac{4}{x + 2} + \frac{2}{x - 2} + \frac{5 x - 6}{4 - x^{2}}$

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Lời giải

$\begin{array}{l} \frac{4}{x + 2} + \frac{2}{x - 2} + \frac{5 x - 6}{4 - x^{2}} = \frac{4}{x + 2} + \frac{2}{x - 2} + \frac{6 - 5 x}{(x - 2) (x + 2)} \\ = \frac{4 (x - 2) + 2 (x + 2) + 6 - 5 x}{(x - 2) (x + 2)} = \frac{4 x - 8 + 2 x + 4 + 6 - 5 x}{(x - 2) (x + 2)} \\ = \frac{x + 2}{(x - 2) (x + 2)} = \frac{1}{x - 2} \end{array}$

Câu 3