( 1 + 1/1 ) x ( 1 + 1/2 ) x ... x ( 1 + 1/n) = 2070
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HT
3
NM
23 tháng 4 2021
Tìm số tự nhiên n, biết rằng ( 1 + 1/1 ) x ( 1 + 1/2 ) x ( 1 + 1/3 ) x ... x ( 1 + 1/n ) = 2070
( 1 + 1/1 ) x ( 1 + 1/2 ) x ( 1 + 1/3 ) x ... x ( 1 + 1/n ) = \(\frac{2}{1}\).\(\frac{3}{2}\).\(\frac{4}{3}\).\(\frac{5}{4}\)....\(\frac{n+1}{n}\)= (n+1)
=> (n+1)=2070
=> n=2069
NV
Nguyễn Việt Lâm
Giáo viên
22 tháng 4 2022
\(S\left(x\right)=\dfrac{1}{x^2}+\dfrac{2}{x^3}+...+\dfrac{n}{x^{n+1}}\)
\(\Rightarrow x.S\left(x\right)=\dfrac{1}{x}+\dfrac{2}{x^2}+\dfrac{3}{x^3}+...+\dfrac{n}{x^n}\)
\(\Rightarrow x.S\left(x\right)-S\left(x\right)=\dfrac{1}{x}+\dfrac{1}{x^2}+\dfrac{1}{x^3}+...+\dfrac{1}{x^n}-\dfrac{n}{x^{n+1}}\)
\(\Rightarrow\left(x-1\right)S\left(x\right)=\dfrac{1}{x}.\dfrac{1-\left(\dfrac{1}{x}\right)^n}{1-\dfrac{1}{x}}-\dfrac{n}{x^{n+1}}=\dfrac{x^n-1}{x^n\left(x-1\right)}-\dfrac{n}{x^{n+1}}=\dfrac{x^{n+1}-x-n\left(x-1\right)}{x^{n+1}\left(x-1\right)}\)
\(\Rightarrow S\left(x\right)=\dfrac{x^{n+1}-\left(n+1\right)x+n}{x^{n+1}\left(x-1\right)^2}\)
1 tháng 7 2018
Bài 1 :
a) \(x\left(x+1\right)\left(x-1\right)-\left(x^2-1\right)\left(x+1\right)\)
\(=\left(x^3-x\right)-\left(x^3+x^2-x-1\right)\)
\(=x^3-x-x^3-x^2+x+1\)
\(=1-x^2\)
b) \(\left(x+1\right)\left(x-2\right)-\left(2x-1\right)\left(x+2\right)+2x\left(x-1\right)\)
\(=\left(x^2-x+2\right)-\left(2x^2+3x-2\right)+\left(2x^2-2x\right)\)
\(=x^2-x+2-2x^3-3x+2+2x^3+2x\)
\(=x^2-2x+4\)
\(=\left(x^2-2x.\dfrac{1}{2}+\dfrac{1}{4}\right)+\dfrac{15}{4}\)
\(=\left(x-\dfrac{1}{2}\right)^2+\dfrac{15}{4}\)
c) \(\left(x^2+2x-1\right)\left(x+2\right)-\left(x-1\right)\left(2x+1\right)\)
\(=\left(x^3+4x^2+3x-2\right)-\left(2x^2-x-1\right)\)
\(=x^3+4x^2+3x-2-2x^3+x+1\)
\(=-x^3+4x^2+4x-1\)
TT
1 tháng 7 2018
Bài 1
\(a)x\left(x+1\right)\left(x-1\right)-\left(x^2-1\right)\left(x+1\right)\\ =\left(x+1\right)\left[x\left(x-1\right)-\left(x^2-1\right)\right]\\ =\left(1+x\right)\left(x^2-x-x^2+1\right)\\ =\left(1+x\right)\left(1-x\right)\\ =1-x^2\)
\(b)\left(x+1\right)\left(x-2\right)-\left(2x-1\right)\left(x+2\right)+2x\left(x-1\right)\\ =x^2-2x+x-2-\left(2x^2+4x-x-2\right)+2x^2-2x\\ =x^2-2x+x-2-(2x^2+3x-2)+2x^2-2x\\ =x^2-2x+x-2-2x^2-3x+2+2x^2-2x\\ =x^2-6x\)
\(c)\left(x^2+2x-1\right)\left(x+2\right)-\left(x-1\right)\left(2x+1\right)\\ =x^3+2x^2+2x^2+4x-x-2-\left(2x^2+x-2x-1\right)\\ =x^3+2x^2+2x^2+4x-x-2-\left(2x^2-x-1\right)\\ =x^3+2x^2+2x^2+4x-x-2-2x^2+x+1\\ =x^3+2x^2+4x-1\)
1 tháng 10 2021
a: Ta có: \(\left(x^2-2x+2\right)\left(x^2-2\right)\left(x^2+2x+2\right)\left(x^2+2\right)\)
\(=\left(x^4-4\right)\left[\left(x^2+2\right)^2-4x^2\right]\)
\(=\left(x^4-4\right)\left(x^4+4x^2+4-4x^2\right)\)
\(=\left(x^4-4\right)\cdot\left(x^4+4\right)\)
\(=x^8-16\)
b: Ta có: \(\left(x+1\right)^2-\left(x-1\right)^2+3x^2-3x\left(x+1\right)\left(x-1\right)\)
\(=x^2+2x+1-x^2+2x-1+3x^2-3x\left(x^2-1\right)\)
\(=3x^2+4x-3x^3+3x\)
\(=-3x^3+3x^2+7x\)
`Answer:`
Ta xét vế trái: \(\left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)...\left(1+\frac{1}{n}\right)\)
\(=2.\frac{3}{2}.\frac{4}{3}...\frac{n+1}{n}\)
\(=\frac{2.3.4...\left(n+1\right)}{2.3...n}\)
\(=n+1\)
\(\Rightarrow n+1=2070\)
\(\Rightarrow n=2070-1\)
\(\Rightarrow n=2069\)