Question

CMR:3^1*3+3^3*5+.....+3^n*(n+2)=3n+3^2n+4 

Với n không thuộc N và n không bằng 0

Answer 1

Sửa đề: \(\frac{3}{1\cdot3}+\frac{3}{3\cdot5}+\cdots+\frac{3}{n\left(n+2\right)}=\frac{3n+3}{2n+4}\)

Ta có: \(\frac{3}{1\cdot3}+\frac{3}{3\cdot5}+\cdots+\frac{3}{n\left(n+2\right)}\)

\(=\frac32\left(\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\cdots+\frac{2}{n\left(n+2\right)}\right)\)

\(=\frac32\left(1-\frac13+\frac13-\frac15+\cdots+\frac{1}{n}-\frac{1}{n+2}\right)=\frac32\left(1-\frac{1}{n+2}\right)\)

\(=\frac32\cdot\frac{n+2-1}{n+2}=\frac{3\left(n+1\right)}{2\left(n+2\right)}=\frac{3n+3}{2n+4}\)