Question

C/m các đẳng thức sau:

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

b) \(\dfrac{2}{a\left(a+1\right)\left(a+2\right)}=\dfrac{1}{a\left(a+1\right)}-\dfrac{1}{\left(a+1\right)\left(a+2\right)}\)

Answer 1

Thực hiện phép tính ở VP ta có:

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

VP bằng VT nên đẳng thức trên là đúng

b) \(\dfrac{1}{a\left(a+1\right)}-\dfrac{1}{\left(a+1\right)\left(a+2\right)}=\dfrac{a+2}{a\left(a+1\right)\left(a+2\right)}-\dfrac{a}{a\left(a+1\right)\left(a+2\right)}\)

\(=\dfrac{2}{a\left(a+1\right)\left(a+2\right)}\)

VP bằng VT nên đẳng thức trên là đúng

Answer 2

a, \(\dfrac{1}{a.\left(a+1\right)}=\dfrac{1}{a}-\dfrac{1}{a+1}\)

Ta có:

\(VP=\dfrac{1}{a}-\dfrac{1}{a+1}=\dfrac{\left(a+1\right)-a}{a\left(a+1\right)}=\dfrac{1}{a\left(a+1\right)}=VT\)

\(\rightarrow\) đpcm

b, \(\dfrac{2}{a\left(a+1\right)\left(a+2\right)}=\dfrac{1}{a\left(a+1\right)}-\dfrac{1}{\left(a+1\right)\left(a+2\right)}\)

Ta có:

\(VP=\dfrac{1}{a\left(a+1\right)}-\dfrac{1}{\left(a+1\right)\left(a+2\right)}=\dfrac{1}{a}-\dfrac{1}{a+1}-\dfrac{1}{a+1}+\dfrac{1}{a+2}\)(áp dụng câu a)

\(=\dfrac{1}{a}-\dfrac{2}{a+1}+\dfrac{1}{a+2}\)

\(=\dfrac{\left(a+1\right)-2a}{a\left(a+1\right)}+\dfrac{1}{a+2}=\dfrac{\left[\left(a+1\right)-2a\right]\left(a+2\right)}{a.\left(a+1\right)\left(a+2\right)}\)

\(=\dfrac{\left(1-a\right)\left(a+2\right)+a\left(a+1\right)}{a\left(a+1\right)\left(a+2\right)}=\dfrac{a+2-a^2-2a+a^2+a}{a\left(a+1\right)\left(a+2\right)}\)

\(=\dfrac{2}{a\left(a+1\right)\left(a+2\right)}=VT\)

Chúc bạn học tốt!!!