Question

Cho biểu thức : \(P\left(x\right)=\dfrac{1}{x^2+x}+\dfrac{1}{x^2+3x+2}+\dfrac{1}{x^2+5x+6}+\dfrac{1}{x^2+7x+12}+\dfrac{1}{x^2+9x+20}\)

a) Tính giá trị của P(\(\dfrac{\sqrt{29}-5}{2}\)); P\(\left(\dfrac{1}{2009}\right)\)

b)Tìm x biết \(P\left(x\right)=\dfrac{5}{4046126}\)

Answer 1

Ta có : \(P=\dfrac{1}{x^2+x}+\dfrac{1}{x^2+3x+2}+\dfrac{1}{x^2+5x+6}+\dfrac{1}{x^2+7x+12}+\dfrac{1}{x^2+9x+20}\)

=\(\dfrac{1}{x\left(x+1\right)}+\dfrac{1}{\left(x+1\right)\left(x+2\right)}+\dfrac{1}{\left(x+2\right)\left(x+3\right)}+\dfrac{1}{\left(x+3\right)\left(x+4\right)}+\dfrac{1}{\left(x+4\right)\left(x+5\right)}\)

=\(\dfrac{1}{x}-\dfrac{1}{x+1}+\dfrac{1}{x+1}-...+\dfrac{1}{x+4}-\dfrac{1}{x+5}\)

= \(\dfrac{1}{x}-\dfrac{1}{x+5}=\dfrac{5}{x\left(x+5\right)}\)

a, Với x=\(\dfrac{\sqrt{29}-5}{2}\Rightarrow A=\dfrac{5}{\dfrac{\sqrt{29}-5}{2}\left(\dfrac{\sqrt{29}-5}{2}+5\right)}\)

Mấy cái còn lại tương tự , bạn tự làm nha