Quy đồng lên rồi tính bình thường thôi bạn

\(\frac{1}{2.5}+\frac{1}{5.8}+...+\frac{1}{\left(3n-1\right)\left(3n+2\right)}\)

\(=\frac{1}{3}.\left(\frac{3}{2.5}+\frac{3}{5.8}+...+\frac{3}{\left(3n-1\right)\left(3n+2\right)}\right)\)

\(=\frac{1}{3}.\left(\frac{1}{2}-\frac{1}{5}+\frac{1}{5}-\frac{1}{8}+...+\frac{1}{3n-1}-\frac{1}{3n+2}\right)\)

\(=\frac{1}{3}.\left(\frac{1}{2}-\frac{1}{3n+2}\right)\)

\(=\frac{1}{3}.\frac{3n}{2.\left(3n+2\right)}\)

\(=\frac{n}{2\left(3n+2\right)}\)

*n=1 thấy: 2=1x4/2 =>* đúng

Giả sử * đúng với n=k, ta có: 2+5+8+...+3k-1=k(3k+1)/2

=> 2+5+8+...+(3k-1)+(3k+2)=k(3k+1)/2+3k+2=(k(3k+1)+6k+4)/2

=> (k(3k+1)+3k+3k+4)/2=(k(3k+4)+3k+4)/2=(k+1)(3k+4)/2

tức là  2+5+8+...+3k+1=(k+1)(3k+4)/2

=> * đúng với n=k+1

=> Theo nguyên lí quy nạp => * đúng với mọi n thuộc N*

Chuyên toán sao học quy nạp sớm thế. 

\(A=\left(1+\frac{2}{4}\right)\left(1+\frac{2}{10}\right)...\left(1+\frac{2}{n^2+3n}\right)\)

\(A=\left(\frac{6}{4}\right)\left(\frac{12}{10}\right)...\left(\frac{n^2+3n+2}{n^2+3n}\right)\)

\(A=\left(\frac{2.3}{1.4}\right)\left(\frac{3.4}{2.5}\right)\left(\frac{4.5}{3.6}\right)...\left(\frac{\left(n+1\right)\left(n+2\right)}{n\left(n+3\right)}\right)\)

\(A=\frac{2.3.4...\left(n+1\right)}{1.2.3...n}.\frac{3.4.5...\left(n+2\right)}{4.5.6...\left(n+3\right)}=\left(n+1\right).\frac{3}{\left(n+3\right)}=\frac{3\left(n+1\right)}{n+3}\)

Do \(0< n+1< n+3\Rightarrow\frac{n+1}{n+3}< 1\Rightarrow\frac{3\left(n+1\right)}{n+3}< 3\)

Vậy \(A< 3\)

a: \(\left(n^2+3n-1\right)\left(n+2\right)-n^3+2\)

\(=n^3+2n^2+3n^2+6n-n-2+n^3+2\)

\(=5n^2+5n=5\left(n^2+n\right)⋮5\)

b: \(\left(6n+1\right)\left(n+5\right)-\left(3n+5\right)\left(2n-1\right)\)

\(=6n^2+30n+n+5-6n^2+3n-10n+5\)

\(=24n+10⋮2\)

d: \(=\left(n+1\right)\left(n^2+2n\right)\)

\(=n\left(n+1\right)\left(n+2\right)⋮6\)

Ngân ơi, bài ai giao thế ?

a,

\(\left(n^2+3n-1\right)\left(n+2\right)-n^3+2\\ =\left(n^2+3n-1\right)n+\left(n^2+3n-1\right)2-n^3+2\\ =n^3+3n^2-n+2n^2+6n-2-n^3+2\\ =5n^2+5n\\ =5\cdot\left(n^2+n\right)⋮5\\ \RightarrowĐpcm\)

b,

\(\left(6n+1\right)\left(n+5\right)-\left(3n+5\right)\left(2n-1\right)\\ =\left(6n+1\right)n+\left(6n+1\right)5-\left(3n+5\right)2n-\left(3n+5\right)\\ =6n^2+n+30n+5-6n^2-10n-3n-5\\ =18n⋮2\\ \RightarrowĐpcm\)