Rút gọn: (1/1*2*3)+(1/2*3*4)+(1/3*4*5)+...+(1/(n-1)*n*(n+1))
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NV
Nguyễn Việt Lâm
Giáo viên
21 tháng 7 2021
\(\dfrac{1}{2!}+\dfrac{2}{3!}+\dfrac{3}{4!}+...+\dfrac{n-1}{n!}\)
\(=\dfrac{2-1}{2!}+\dfrac{3-1}{3!}+\dfrac{4-1}{4!}+...+\dfrac{n-1}{n!}\)
\(=\dfrac{2}{2!}-\dfrac{1}{2!}+\dfrac{3}{3!}-\dfrac{1}{3!}+...+\dfrac{n}{n!}-\dfrac{1}{n!}\)
\(=1-\dfrac{1}{2!}+\dfrac{1}{2!}-\dfrac{1}{3!}+...+\dfrac{1}{\left(n-1\right)!}-\dfrac{1}{n!}\)
\(=1-\dfrac{1}{n!}\)
15 tháng 7 2019
1)\(n^2\left(n-1\right)\left(n+1\right)-\left(n^2+2\right)\left(n^2-2\right)=n^2\left(n^2-1\right)-\left(n^4-4\right)=n^4-n^2-n^4+4\)
\(=-n^2+4\)
2)\(\left(y+3\right)\left(y-3\right)\left(y^2+9\right)-\left(y^2-4\right)\left(y^2+4\right)=\left(y^2-9\right)\left(y^2+9\right)-\left(y^4-16\right)\)
\(=y^4-81-y^4+16=-65\)
3)\(\left(x-2y+3\right)\left(x+2y-3\right)-\left(x-2y\right)\left(x+2y\right)=\left(x+3\right)^2-4y^2-\left(x^2-4y^2\right)\)
\(=x^2+6x+9-4y^2-x^2+4y^2=6x+9\)
4)\(\left(a+b+c\right)^2=a^2+b^2+c^2+2ab+2bc+2ac\)
5)\(\left(a+b-c\right)^2=a^2+b^2+c^2+2ab-2bc-2ac\)
6)\(\left(a-b-c\right)^2=a^2+b^2+c^2-2ab+2bc-2ac\)
Học tốt nha bạn !
21 tháng 11 2016
\(C=\frac{1^2}{2^2-1}.\frac{3^2}{4^2-1}.\frac{5^2}{6^2-1}....\frac{n^2}{\left(n+1\right)^2-1}\)
\(=\frac{1^2}{1.3}.\frac{3^2}{3.5}.\frac{5^2}{5.7}.....\frac{n^2}{n.\left(n+2\right)}\)
\(=\frac{1}{n+2}\)
Đặt \(A=\dfrac{1}{1\cdot2\cdot3}+\dfrac{1}{2\cdot3\cdot4}+...+\dfrac{1}{\left(n-1\right)n\left(n+1\right)}\)
\(2A=\dfrac{2}{1\cdot2\cdot3}+\dfrac{2}{2\cdot3\cdot4}+...+\dfrac{2}{\left(n-1\right)n\left(n+1\right)}\)
\(2A=\dfrac{1}{1\cdot2}-\dfrac{1}{2\cdot3}+\dfrac{1}{2\cdot3}-\dfrac{1}{3\cdot4}+...+\dfrac{1}{\left(n-1\right)n}-\dfrac{1}{n\left(n+1\right)}\)
\(2A=\dfrac{1}{2}-\dfrac{1}{n\left(n+1\right)}\)
\(2A=\dfrac{n\left(n+1\right)-2}{2n\left(n+1\right)}\)
\(A=\dfrac{n\left(n+1\right)-2}{4n\left(n+1\right)}\)