Question

CMR với \(m\in N\):

a)     \(\frac{4}{4m+2}=\frac{1}{m+1}+\frac{1}{\left(m+1\right)\left(2m+1\right)}\)

b)       \(\frac{4}{m+3}=\frac{1}{m+2}+\frac{1}{\left(m+1\right)\left(m+2\right)+}+\frac{1}{\left(m+1\right)\left(4m+3\right)}\)

 

Answer 1

a) \(\frac{1}{m+1}+\frac{1}{\left(m+1\right)\left(2m+1\right)}\)

\(=\frac{2m+1}{\left(m+1\right)\left(2m+1\right)}+\frac{1}{\left(m+1\right)\left(2m+1\right)}\)

\(=\frac{2m+2}{\left(m+1\right)\left(2m+1\right)}\)

\(=\frac{2\left(m+1\right)}{\left(m+1\right)\left(2m+1\right)}\)

\(=\frac{2}{2m+1}=\frac{4}{4m+2}\left(đpcm\right)\)

Answer 2

b) \(\frac{1}{m+2}+\frac{1}{\left(m+1\right)\left(m+2\right)}+\frac{1}{\left(m+1\right)\left(4m+3\right)}\)

\(=\frac{m+1}{\left(m+1\right)\left(m+2\right)}+\frac{1}{\left(m+1\right)\left(m+2\right)}+\frac{1}{\left(m+1\right)\left(4m+3\right)}\)

\(=\frac{m+2}{\left(m+1\right)\left(m+2\right)}+\frac{1}{\left(m+1\right)\left(4m+3\right)}\)

\(=\frac{1}{m+1}+\frac{1}{\left(m+1\right)\left(4m+3\right)}\)

\(=\frac{4m+3}{\left(m+1\right)\left(4m+3\right)}+\frac{1}{\left(m+1\right)\left(4m+3\right)}\)

\(=\frac{4m+4}{\left(m+1\right)\left(4m+3\right)}\)

\(=\frac{4\left(m+1\right)}{\left(m+1\right)\left(4m+3\right)}\)

\(=\frac{4}{4m+3}\left(đpcm\right)\)