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20 tháng 2 2020
Đặt \(B=\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+...+\frac{1}{2014^2}\)
Ta có : \(\frac{1}{3^2}< \frac{1}{2.3}\)
\(\frac{1}{4^2}< \frac{1}{3.4}\)
\(\frac{1}{5^2}< \frac{1}{4.5}\)
...
\(\frac{1}{2014^2}< \frac{1}{2013.2014}\)
\(\Rightarrow B< \frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{2013.2014}\)
\(\Rightarrow B< \frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{2013}-\frac{1}{2014}\)
\(\Rightarrow B< \frac{1}{2}-\frac{1}{2014}< \frac{1}{2}\)
\(\Rightarrow A< \frac{1}{2^2}+\frac{1}{2}=\frac{3}{4}\)
Vậy A<\(\frac{3}{4}\)
20 tháng 2 2020
A<\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2013.2014}\)=\(\frac{2013}{2014}\)<\(\frac{3}{4}\)
HD
6 tháng 3 2020
k chép đề
3/2.A=\(\frac{3}{4}+\left(\frac{3}{2}\right)^2+\left(\frac{3}{2}\right)^3+\left(\frac{3}{2}\right)^4+\left(\frac{3}{2}\right)^5+...+\left(\frac{3}{2}\right)^{2013}\)
3/2A-A=(\(\frac{3}{4}+\left(\frac{3}{2}\right)^2+\left(\frac{3}{2}\right)^3+\left(\frac{3}{2}\right)^4+\left(\frac{3}{2}\right)^5+...+\left(\frac{3}{2}\right)^{2013}\)) - (\(\frac{1}{2}+\frac{3}{2}+\left(\frac{3}{2}\right)^2+\left(\frac{3}{2}\right)^3+\left(\frac{3}{2}\right)^4+...+\left(\frac{3}{2}\right)^{2012}\))
1/2 . A =\(\frac{1}{2}+\left(\frac{3}{2}\right)^{2013}\)
A=\(\frac{\frac{1}{2}+\left(\frac{3}{2}\right)^{2013}}{2}\)
B-A=\(\frac{\left(\frac{3}{2}\right)^{2018}}{2}-\)\(\frac{\frac{1}{2}+\left(\frac{3}{2}\right)^{2013}}{2}\)
\(B-A=\frac{\frac{1}{2}}{2}=\frac{1}{2}:2=\frac{1}{4}\)
XO
31 tháng 8 2020
1) \(\frac{3^{2014}.8^{19}}{6^{60}.3^{1955}}=\frac{3^{2014}.\left(2^3\right)^{19}}{\left(2.3\right)^{60}.3^{1955}}=\frac{3^{2014}.2^{57}}{2^{60}.3^{2015}}=\frac{1}{2^3.3}=\frac{1}{24}\)
2) \(5^x+5^{x+1}=150\)
=> 5x(1 + 5) = 150
=> 5x.6 = 150
=> 5x = 25
=> \(x=\pm2\)
3) \(\frac{3}{11.16}+\frac{3}{16.21}+...+\frac{3}{61.66}=\frac{3}{5}\left(\frac{5}{11.16}+\frac{5}{16.21}+...+\frac{5}{61.66}\right)\)
\(=\frac{3}{5}\left(\frac{1}{11}-\frac{1}{16}+\frac{1}{16}-\frac{1}{21}+...+\frac{1}{61}-\frac{1}{66}\right)=\frac{3}{5}.\left(\frac{1}{11}-\frac{1}{66}\right)=\frac{3}{5}.\frac{5}{66}=\frac{1}{22}\)
4 tháng 10 2018
P = 1+1/2x2+1/3x3+...+1/2014x2014.
Mà: 1/2x2 bé hơn 1/1x2; 1/3x3 bé hơn 1/2x3; 1/2014x2014 bé hơn 1/2013x2014.
P = 1+1/2x2+1/3x3+...+1/2014x2014 bé hơn 1+1/1x2+1/2x3+...+1/2013x2014 = 1+1-1/2+1/2-1/3+...+1/2013-1/2014 = 1+1-1/2014 = 4027/2014; Q = 7/4.(Bạn tự tính nhá)
Suy ra P lớn hơn Q.
13 tháng 12 2018
\(a)A=\frac{24\cdot47-23}{24+47-23}\cdot\frac{3+\frac{3}{7}+\frac{3}{11}+\frac{3}{1001}+\frac{3}{13}}{\frac{9}{1001}+\frac{9}{13}+\frac{9}{7}+\frac{9}{11}+9}\)
\(=\frac{(23+1)\cdot47-23}{24+47-23}\cdot\frac{3+\frac{3}{7}+\frac{3}{11}+\frac{3}{1001}+\frac{3}{13}}{\frac{9}{1001}+\frac{9}{13}+\frac{9}{7}+\frac{9}{11}+9}=\frac{47-23+24}{47-23+24}\cdot\frac{3(1+\frac{1}{7}+\frac{1}{11}+\frac{1}{1001}+\frac{1}{13})}{3(3+\frac{3}{1001}+\frac{3}{13}+\frac{3}{7}+\frac{3}{11})}\)
\(=\frac{1+\frac{1}{7}+\frac{1}{11}+\frac{1}{1001}+\frac{1}{13}}{3+\frac{3}{1001}+\frac{3}{13}+\frac{3}{7}+\frac{3}{11}}=\frac{1+\frac{1}{1001}+\frac{1}{13}+\frac{1}{7}+\frac{1}{11}}{3(1+\frac{1}{1001}+\frac{1}{13}+\frac{1}{7}+\frac{1}{11})}=\frac{1}{3}\)
\(b)\)\(\text{Đặt A = }1+2+2^2+2^3+...+2^{2012}\)
\(2A=2(1+2^2+2^3+...+2^{2012})\)
\(2A=2+2^2+2^3+...+2^{2013}\)
\(2A-A=(2+2^2+2^3+2^4+...+2^{2013})-(1+2+2^2+2^3+...+2^{2012})\)
\(\Rightarrow A=2^{2013}-1\)
\(\text{Quay lại bài toán,ta có :}\)
\(B=\frac{1+2+2^2+2^3+...+2^{2012}}{2^{2014}-2}=\frac{2^{2013}-1}{2^{2014}-2}=\frac{2^{2013}-1}{2(2^{2013}-1)}=\frac{1}{2}\)
Giải
Đặt A=1+2^2+2^3+...+2^2012
2A=2×(1+2+2^2+.....+2^2012)
2A=2+2^2+2^3+2^4+.....+2^2013
2A-A=(2+2^2+2^3+2^4+....+2^2013)-(1+2^2+2^3+2^4+....+2^2012)
\(\Rightarrow\)A=2^2013-1
Ta có : M=1+2+2^2+....+2^2012/2^2014-2
= 2^2013-1/2^2014
= 2^2013-1/2×(2^2013-1)
=1/2
Chúc bạn học tốt
#)Giải :
Gọi phần tử số là A
A = 1 + 2 + 22+ 23+ ... + 22012
2A = 2 = 22 + 23 + ... + 22013
2A - A = ( 2 + 22 + 23 + ... + 22013 ) - ( 1 + 2 + 22 + 23 + ... + 22012)
2A - A = 22013 - 1
=> A = 22013 - 1
Quay trở lại vs baì toán :
\(M=\frac{1+2+2^2+2^3+...+2^{2012}}{2^{2014}-2}\)
\(M=\frac{2^{2013}-1}{2.2^{2013}-2}\)
\(M=\frac{2^{2013}-1}{2\left(2^{2013}-1\right)}=\frac{1}{2}\)
Vậy : \(M=\frac{1}{2}\)
#)Chúc bn học tốt :D