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KN
23 tháng 4 2019
Đặt \(A=\frac{4}{3.5}+\frac{4}{5.7}+\frac{4}{7.9}+...+\frac{4}{99.101}\)
\(\Rightarrow\frac{1}{2}A=\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+...+\frac{2}{99.101}\)
\(\Rightarrow\frac{1}{2}A=\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{99}-\frac{1}{101}\)
\(\Rightarrow\frac{1}{2}A=\frac{1}{3}-\frac{1}{101}\)
\(\Rightarrow\frac{1}{2}A=\frac{101}{303}-\frac{3}{303}\)
\(\Rightarrow\frac{1}{2}A=\frac{98}{303}\)
\(\Rightarrow A=\frac{98}{303}\div\frac{1}{2}\)
\(\Rightarrow A=\frac{199}{303}\)
KN
23 tháng 4 2019
\(x+\frac{3}{22}=\frac{27}{121}.\frac{11}{9}\)
\(\Leftrightarrow x+\frac{3}{22}=\frac{27.11}{121.9}\)
\(\Leftrightarrow x+\frac{3}{22}=\frac{3.1}{11.1}\)
\(\Leftrightarrow x+\frac{3}{22}=\frac{3}{11}\)
\(\Leftrightarrow x=\frac{3}{11}-\frac{3}{22}\)
\(\Leftrightarrow x=\frac{6}{22}-\frac{3}{22}\)
\(\Leftrightarrow x=\frac{3}{22}\)
1 tháng 4 2017
\(\frac{4}{3\cdot5}+\frac{4}{5\cdot7}+\frac{4}{7\cdot9}+...+\frac{4}{97\cdot99}\)
\(=\frac{2\cdot2}{3\cdot5}+\frac{2\cdot2}{5\cdot7}+\frac{2\cdot2}{7\cdot9}+...+\frac{2\cdot2}{97\cdot99}\)
\(=\frac{2}{3}+\frac{2}{5}-\frac{2}{5}+\frac{2}{7}-\frac{2}{7}+\frac{2}{9}-...+\frac{2}{97}-\frac{2}{99}\)
\(=\frac{2}{3}-\frac{2}{99}\)
\(=\frac{64}{99}\)
NT
2
NN
Nguyễn Ngọc Anh Minh
CTVHS
VIP
8 tháng 8 2023
a/
\(A=3^2+3^2.2^2+3^2.3^2+3^2.4^2+...+3^2.30^2=\)
\(=3^2\left(1^2+2^2+3^2+...+30^2\right)\)
Đăt biểu thức trong dấu ngoặc là B
\(B=1\left(2-1\right)+2\left(3-1\right)+3\left(4-1\right)+...+30\left(31-1\right)=\)
\(=1.2+2.3+3.4+30.31-\left(1+2+3+...+30\right)=\)
\(C=1+2+3+...+30=\dfrac{30\left(1+30\right)}{2}=465\)
\(D=1.2+2.3+3.4+...+30.31\)
\(3D=1.2.3+2.3.3+3.4.3+...+30.31.3=\)
\(=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+30.31.\left(32-29\right)=\)
\(=1.2.3-1.2.3+2.3.4-2.3.4+3.4.5-...-29.30.31+30.31.32=\)
\(=30.31.32\Rightarrow D=\dfrac{30.31.32}{3}=10.31.32\)
\(\Rightarrow A=3^2\left(C-D\right)=3^2\left(10.31.32-465\right)\)
b/
Đặt biểu thức là A
\(\dfrac{A}{2}=\dfrac{5-3}{3.5}+\dfrac{7-5}{5.7}+\dfrac{9-7}{7.9}+...+\dfrac{39-37}{37.39}=\)
\(=\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+\dfrac{1}{37}-\dfrac{1}{39}=\)
\(=\dfrac{1}{3}-\dfrac{1}{39}=\dfrac{12}{39}\Rightarrow A=\dfrac{2.12}{39}=\dfrac{24}{39}=\dfrac{8}{13}\)
ND
Nguyễn Đức Trí
VIP
8 tháng 8 2023
\(A=3.3+6.6+9.9+...+90.90\)
\(A=3^2\left(1+2^2+3^2+...+10^2\right)\)
\(A=9.\dfrac{10.\left(10+1\right)\left(2.10+1\right)}{6}\)
\(A=3.\dfrac{10.11.21}{2}\)
\(A=3465\)
NH
3
N
26 tháng 4 2017
\(\frac{4}{1.3}\)+\(\frac{4}{3.5}\)+\(\frac{4}{5.7}\)+\(\frac{4}{7.9}\)+...+\(\frac{4}{2011.2013}\)
= 1+\(\frac{1}{3}\)-\(\frac{1}{3}\)+\(\frac{1}{5}\)-\(\frac{1}{5}\)+\(\frac{1}{7}\)-\(\frac{1}{7}\)+\(\frac{1}{9}\)+...+\(\frac{1}{2011}\)+\(\frac{1}{2013}\)
=1+ 0 + 0 + 0 +...+ 0 + \(\frac{1}{2013}\)
=1+\(\frac{1}{2013}\)
=\(\frac{2014}{2013}\)
k dùm nha
MV
26 tháng 4 2017
\(\frac{4}{1\cdot3}+\frac{4}{3\cdot5}+\frac{4}{5\cdot7}+...+\frac{4}{2011\cdot2013}\)
\(=2\cdot\left(\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\frac{2}{5\cdot7}+...+\frac{2}{2011\cdot2013}\right)\)
\(=2\cdot\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{2011}-\frac{1}{2013}\right)\)
\(=2\cdot\left(1-\frac{1}{2013}\right)\)
\(=2\cdot\frac{2012}{2013}\)
\(=\frac{4024}{2013}\)
NT
2
NH
1
2
7 tháng 5 2015
\(\frac{4}{1.3}+\frac{4}{3.5}+...+\frac{4}{2013.2015}=2.\left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{2013.2015}\right)=2.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{2013}-\frac{1}{2015}\right)\)
\(=2.\left(\frac{2015}{2015}-\frac{1}{2015}\right)\)
\(=2.\frac{2014}{2015}\)
\(=\frac{4028}{2015}\)
LT
0
LS
2
26 tháng 4 2022
\(B=\dfrac{2}{3.5}+\dfrac{2}{5.7}+\dfrac{2}{7.9}+...+\dfrac{2}{99.100}\)
\(B=\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{9}+...+\dfrac{1}{97}-\dfrac{1}{99}+\dfrac{1}{99.100}+\dfrac{1}{99.100}\)
\(B=\dfrac{1}{3}-\dfrac{1}{99}+\dfrac{1}{99}-\dfrac{1}{100}+\dfrac{1}{99}-\dfrac{1}{100}\)
\(B=\dfrac{1}{3}-\dfrac{2}{100}+\dfrac{1}{99}\)
\(B=\dfrac{1}{3}-\dfrac{1}{50}+\dfrac{1}{99}\)
Đến đây thì hết tính hợp lý được rồi:v
\(B=\dfrac{34}{99}-\dfrac{1}{50}\)
\(B=\dfrac{1601}{4950}\)
4/3.5+4/5.7+...+4/98.100(sửa vì đề sai)
=2(1/3-1/5+1/5-1/7+...+1/98-1/100)
=2(1/3-1/100)
=2.(97/300)
=97/150
\(\frac{4}{3.5}+\frac{4}{5.7}+\frac{4}{7.9}+...+\frac{4}{99.100}\)
\(=2.\left(\frac{2}{3.5}+\frac{2}{5.7}+\frac{2}{7.9}+...+\frac{2}{99.100}\right)\)
\(=2.\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{99}-\frac{1}{100}\right)\)
\(=2.\left(\frac{1}{3}-\frac{1}{100}\right)\)
\(=2.\frac{97}{300}\)
\(=\frac{97}{150}\)