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\(\left[9-\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{10}\right)\right]\div\left(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...+\frac{9}{10}\right)\)
\(=\left[\left(1+1+1+...+1\right)-\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{10}\right)\right]\div\left(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...+\frac{9}{10}\right)\)
có 9 số 1 có 9 số hạng
\(=\left[\left(1-\frac{1}{2}\right)+\left(1-\frac{1}{3}\right)+\left(1-\frac{1}{4}\right)+...+\left(1-\frac{1}{10}\right)\right]\div\left(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...+\frac{9}{10}\right)\)
\(=\left[\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...+\frac{9}{10}\right]\div\left(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...+\frac{9}{10}\right)\)
\(=1\)
[ 212 + ( 48 - 2 . x . x2 ) : 40 - 3 = 3
[ 212 + ( 48 - 2 . x . x2 ) : 40 = 3 + 3 = 6
[ 212 + ( 48 - 2 . x . x2 ) = 6 . 40 = 240
=> 48 - 2 . x . x2 = 240 - 212 = 28
2 . x . x2 = 48 - 28 = 20
x. x2 = 20 : 2 = 10
Đặt \(x=\frac{y}{2}=\frac{z}{3}=k\left(k\in Q\right)\)\(\Rightarrow x=k;y=2k;z=3k\)
Thế (1) vào biểu thức trên
\(\Rightarrow2\left(x^2+y^2\right)-z^2=9\)
\(\Leftrightarrow2\left[\left(k\right)^2+\left(2k\right)^2\right]-\left(3k\right)^2=9\)
\(\Rightarrow2\left(k^2+4k^2\right)-9k^2=9\)
\(\Rightarrow2k^2+8k^2-9k^2=9\)
\(\Rightarrow k^2=9\)
\(\Rightarrow k=\hept{\begin{cases}3\\-3\end{cases}}\)
Với k = 3
\(\Rightarrow x=3;y=3.2=6;z=3.3=9\)
Với k = -3
\(\Rightarrow x=-3;y=-3.2=-6;z=-3.3=-9\)
Đặt \(A=1.2+2.3+3.4+...+n\left(n+1\right)\)
\(\Rightarrow3A=1.2.3+2.3.3+3.4.3+...+3n\left(n+1\right)\)
\(=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+n\left(n+1\right)\left(n+2-n+1\right)\)
\(=1.2.3+2.3.4-1.2.3+...+n\left(n+1\right)\left(n+2\right)-\left(n-1\right)n\left(n+1\right)\)
\(=n\left(n+1\right)\left(n+2\right)\)
\(\Rightarrow1.2+2.3+3.4+...+n\left(n+1\right)=\frac{n\left(n+1\right)\left(n+2\right)}{3}\)
\(\Rightarrow S=1-\frac{1}{4}+\frac{1}{4}-\frac{1}{9}+..+\frac{1}{n^2}-\frac{1}{n+1^2}\)
\(\Rightarrow S=1-\frac{1}{n+1}\)
\(\Rightarrow S+\frac{n}{n+1}\)