Question

Tính nhanh giá trị biểu thức sau :

A= \(\frac{1}{\left(x+1\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)+\left(x+5\right)}+\frac{1}{\left(x+5\right)\left(x+7\right)}+\frac{1}{\left(x+7\right)\left(x+9\right)}+\frac{1}{\left(x+9\right)\left(x+11\right)}\)

Answer 1

 \(A=\frac{1}{\left(x+1\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+5\right)}+\frac{1}{\left(x+5\right)\left(x+7\right)}+\frac{1}{\left(x+7\right)\left(x+9\right)}+\frac{1}{\left(x+9\right)\left(x+11\right)}\)

\(=\frac{1+1+1+1+1}{\left(x+1\right)\left(x+3\right)\left(x+5\right)\left(x+7\right)\left(x+9\right)\left(x+11\right)}\)

\(=\frac{5}{\left(x+1\right)\left(x+3\right)\left(x+5\right)\left(x+7\right)\left(x+9\right)\left(x+11\right)}\)

\(=\frac{5}{\left(x+1\right)\left(x+11\right)\left(x+3\right)\left(x+9\right)\left(x+5\right)\left(x+7\right)}\)

\(=\frac{5}{\left(x^2+11x+x+11\right)\left(x^2+9x+3x+27\right)\left(x^2+7x+5x+35\right)}\)

\(=\frac{5}{\left(x^2+12x+11\right)\left(x^2+12x+27\right)\left(x^2+12x+35\right)}\)

Answer 2

A=\(\frac{1}{x+1}-\frac{1}{x+3}+\frac{1}{x+3}-\frac{1}{x+5}+\frac{1}{x+5}-\frac{1}{x+7}+\frac{1}{x+7}-\frac{1}{x+9}+\frac{1}{x+9}-\frac{1}{x+11}\)

Rút gọn hết đi ta có \(\frac{1}{x+1}-\frac{1}{x+11}\)=\(\frac{x+11}{\left(x+1\right).\left(x+11\right)}-\frac{x+1}{\left(x+1\right).\left(x+11\right)}\)

A=\(\frac{x+11-x-1}{\left(x+1\right).\left(x+11\right)}\)

A=\(\frac{10}{x^2+12x+11}\)

Answer 3

\(A=\frac{1}{\left(x+1\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+5\right)}+\frac{1}{\left(x+5\right)\left(x+7\right)}+\frac{1}{\left(x+7\right)\left(x+9\right)}+\frac{1}{\left(x+9\right)\left(x+11\right)}\)

\(A=\frac{1}{x+1}-\frac{1}{x+3}+\frac{1}{x+3}-\frac{1}{x+5}+...+\frac{1}{x+9}-\frac{1}{x+11}\)

\(A=\frac{1}{x+1}-\frac{1}{x+11}\Leftrightarrow A=\frac{x+11}{\left(x+1\right)\left(x+11\right)}+\frac{x+1}{\left(x+1\right)\left(x+11\right)}\)

\(\Leftrightarrow A=\frac{2x+12}{\left(x+1\right)\left(x+11\right)}\)

Answer 4

 A=1(x+1)(x+3) +1(x+3)(x+5) +1(x+5)(x+7) +1(x+7)(x+9) +1(x+9)(x+11) 

=1+1+1+1+1(x+1)(x+3)(x+5)(x+7)(x+9)(x+11) 

=5(x+1)(x+3)(x+5)(x+7)(x+9)(x+11) 

=5(x+1)(x+11)(x+3)(x+9)(x+5)(x+7) 

=5(x2+11x+x+11)(x2+9x+3x+27)(x2+7x+5x+35) 

=5(x2+12x+11)(x2+12x+27)(x2+12x+35) 

Answer 5

\(A=\frac{1}{\left(x+1\right)\left(x+3\right)}+\frac{1}{\left(x+5\right)\left(x+7\right)}+\frac{1}{\left(x+7\right)\left(x+9\right)}+\frac{1}{\left(x+9\right)\left(x+11\right)}\)

=> 2A=\(\frac{2}{\left(x+1\right)\left(x+3\right)}+\frac{2}{\left(x+3\right)\left(x+5\right)}+\frac{2}{\left(x+5\right)\left(x+7\right)}+\frac{2}{\left(x+7\right)\left(x+9\right)}+\frac{2}{\left(x+9\right)\left(x+11\right)}\)

=> 2A=\(\frac{1}{x+1}-\frac{1}{x+3}+\frac{1}{x+3}-\frac{1}{x+5}+....-\frac{1}{x+11}\)

=> 2A=\(\frac{1}{x+1}-\frac{1}{x+11}=\frac{x+11}{\left(x+1\right)\left(x+11\right)}-\frac{x+1}{\left(x+1\right)\left(x+11\right)}=\frac{10}{\left(x+1\right)\left(x+11\right)}\)

=> A=\(\frac{10}{\left(x+1\right)\left(x+11\right)}:2=\frac{10}{\left(x+1\right)\left(x+11\right)}\cdot\frac{1}{2}=\frac{5}{\left(x+1\right)\left(x+11\right)}\)

Answer 6

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Answer 7

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