2016/1.3 + 2016/3.5 + 2016/5.7 +...+ 2016/2015.2017=?
B=4/3.5-6/5.7+8/7.9-10/9.11+.....+2016/2015.2017-2018/2017.2019
4\3.5 .6\5.7......2016\2015.2017
Cm tổng trên >1\6
tính A=\(\frac{2^2}{1.3}+\frac{3^2}{2.4}+\frac{4^2}{3.5}+...+\frac{2016^2}{2015.2017}\)
A=4/3+9/8+16/15+..............+4064256/4064255
A=1+1/3+1+1/8+1/15+...............+1/4064255
A=(1+1+...+1)+(1/3+1/8+...+1/406255) (có 2015 số 1)
A=2015+(1/1.3+1/2.4+...........+1/2015.2017)
A=2015+1/2(1/1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+1/5-1/7+....+1/2012-1/2014+1/2013-1/2015+1/2014-1/2016+1/2015-1/2017)
A=2015+1/2(1+1/2-1/2016-1/2017)
A=2015,749504
k cho mình nhé mình k lại cho
\(M=\left(\dfrac{4}{1.3}-\dfrac{8}{3.5}+\dfrac{12}{5.7}-...+\dfrac{4028}{2013.2015}\right).\dfrac{2015}{2016}\)
Chứng minh M là số tự nhiên.
Trong dấu ngoặc đơn có số các số hạng là
Đặt tổng các số hạng trong ngoặc đơn là A
\(\dfrac{2013-1}{2}+1=1007\) số hạng
\(A=\dfrac{3+1}{1.3}-\dfrac{5+3}{3.5}+\dfrac{7+5}{5.7}-...+\dfrac{2015+2013}{2013.2015}=\)
\(=1+\dfrac{1}{3}-\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}+\dfrac{1}{7}-...+\dfrac{1}{2013}+\dfrac{1}{2015}=1+\dfrac{1}{2015}=\dfrac{2016}{2015}\)
\(\Rightarrow M=A.\dfrac{2015}{2016}=\dfrac{2016}{2015}.\dfrac{2015}{2016}=1\) là số tự nhiên
Tính tổng: \(\dfrac{1^2}{1.3}+\dfrac{2^2}{3.5}+\dfrac{3^2}{5.7}+...+\dfrac{2016^2}{4031.4033}\)
\(\sum\limits^{2016}_{x=1}\left(\dfrac{x^2}{\left(2x-1\right)\left(2x+1\right)}\right)\)
Chứng minh rằng: 1.3+2/2^2+2.4+2/3^2+3.5+2/4^2+...+2010.2012+2/2011^2+2015.2017+2/2016^2<2017
Mình cần gấp lắm, cố xong trong hôm nay nha
Đặt A = \(\frac{1.3+2}{2^2}+\frac{2.4+2}{3^2}+\frac{3.5+2}{4^2}+...+\frac{2010.2012+2}{2011^2}+\frac{2015.2017+2}{2016^2}\)
\(=\frac{\left(2-1\right)\left(2+1\right)+2}{2^2}+\frac{\left(3-1\right)\left(3+1\right)}{3^2}+...+\frac{\left(2016-1\right)\left(2016+1\right)+2}{2016^2}\)
\(=\frac{2^2-1+2}{2^2}+\frac{3^2-1+2}{3^2}+....+\frac{2016^2-1+2}{2016^2}\)
\(=\frac{2^2+1}{2^2}+\frac{3^2+1}{3^2}+...+\frac{2016^2+1}{2016^2}\)
\(=\left(1+1+...+1\right)+\left(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2016^2}\right)\)(2015 hạng tử 1)
\(=2015+\left(\frac{1}{2.2}+\frac{1}{3.3}+...+\frac{1}{2016.2016}\right)\)
\(< 2015+\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2015.2016}\right)\)
\(=2015+\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}+...+\frac{1}{2015}-\frac{1}{2016}\right)=2015+\left(1-\frac{1}{2016}\right)\)
= 2015 + 1 + 1/2016
= 2016 + 1/2016 < 2017
=> A < 2017 (ĐPCM)
Tính:
\(a,A=\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{2015.2017}\)
\(b.B=\frac{-1}{3}+\frac{1}{3^2}-\frac{1}{3^3}+...+\frac{1}{3^{2016}}\)
Tìm n biết
\(\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+.....+\frac{2}{n\left(n+2\right)}<\frac{2015}{2016}\)
<=>2-2/3+2/3-2/5........+2n-2n+2<2015/2016
<=>2-2n+2<2015/2016
=>n+2=1/2016
=>n=2014
\(\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+......+\frac{2}{n\left(n+2\right)}\)<\(\frac{2015}{2016}\)
VT=\(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+....+\frac{1}{5}-\frac{1}{n+2}\)=\(1-\frac{1}{n+2}\)
Ta có:\(1-\frac{1}{n+2}=\frac{2015}{2016}\Rightarrow\)\(\frac{1}{n+2}=1-\frac{2015}{2016}\)
\(\Rightarrow\)\(\frac{1}{n+2}=\frac{1}{2016}=n+2=2016\)
\(\Rightarrow\)\(n=2014\)
Vậy\(n=2014\)
bài 1:tìm x:
2/1.3+2/3.5+2/5.7+.........+2/x(x+2)=2015/2016
đó là phân số nhé
Bài này dễ ý mà, vô cùng đơn giản..........
Ta có:
\(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{x\left(x+2\right)}=\dfrac{2015}{2016}.\)
\(\dfrac{2}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{x}-\dfrac{1}{x+2}\right)=\dfrac{2015}{2016}.\)
\(1\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{x}-\dfrac{1}{x+2}\right)=\dfrac{2015}{2016}.\)
\(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{x}-\dfrac{1}{x+2}=\dfrac{2015}{2016}.\)
\(\left(\dfrac{1}{3}-\dfrac{1}{3}\right)+\left(\dfrac{1}{5}-\dfrac{1}{5}\right)+...+\left(\dfrac{1}{x}-\dfrac{1}{x}\right)+\left(1-\dfrac{1}{x+2}\right)=\dfrac{2015}{2016}.\)
\(0+0+...+0+\left(1-\dfrac{1}{x+2}\right)=\dfrac{2015}{2016}.\)
\(1-\dfrac{1}{x+2}=\dfrac{2015}{2016}.\)
\(\dfrac{1}{x+2}=1-\dfrac{2015}{2016}.\)
\(\dfrac{1}{x+2}=\dfrac{1}{2016}.\)
\(\Rightarrow x+2=2016.\)
\(\Rightarrow x=2016-2=2014.\)
Vậy \(x=2014.\)
~ Học tốt nha bn!!! ~
Bài mik đúng thì nhớ tick mik nha!!!