1/1x3+1/3x5+1/5x7+...+1/(2n+1)x(2n+3)=n+1/2n+3
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VT
1
TM
10 tháng 7 2016
\(\frac{1}{1x3}+\frac{1}{3x5}+\frac{1}{5x7}+...+\frac{1}{\left(2n+1\right)x\left(2x+3\right)}=\frac{n+1}{2n+3}\)
=>\(2x\left(\frac{1}{1x3}+\frac{1}{3x5}+\frac{1}{5x7}+...+\frac{1}{\left(2n+1\right)x\left(2n+3\right)}\right)=2x\frac{n+1}{2n+3}\)
=>\(\frac{2}{1x3}+\frac{2}{3x5}+\frac{2}{5x7}+...+\frac{2}{\left(2n+1\right)\left(2n+3\right)}=\frac{2n+2}{2n+3}\)
=>\(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{2n+1}-\frac{1}{2n+3}=\frac{2n+2}{2n+3}\)
=>\(1-\frac{1}{2n+3}=\frac{2n+2}{2n+3}\)
=>\(\frac{2n+2}{2n+3}=\frac{2n+2}{2n+3}\)
=>.....
17 tháng 12 2017
a, Đặt :
\(A=\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+..............+\dfrac{1}{19.21}\)
\(\Leftrightarrow2A=\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+............+\dfrac{2}{19.21}\)
\(\Leftrightarrow2A=1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+..........+\dfrac{1}{19}-\dfrac{1}{21}\)
\(\Leftrightarrow2A=1-\dfrac{1}{21}\)
\(\Leftrightarrow2A=\dfrac{20}{21}\)
\(\Leftrightarrow A=\dfrac{10}{21}\)
17 tháng 12 2017
b, \(A=\dfrac{1}{1.3}+\dfrac{1}{3.5}+...........+\dfrac{1}{\left(2n-1\right)\left(2n+1\right)}\)
\(\Leftrightarrow2A=\dfrac{2}{1.3}+\dfrac{2}{3.5}+............+\dfrac{2}{\left(2n-1\right)\left(2n+1\right)}\)
\(\Leftrightarrow2A=1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+........+\dfrac{1}{2n-1}-\dfrac{1}{2n+1}\)
\(\Leftrightarrow2A=1-\dfrac{1}{2n+1}\)
\(\Leftrightarrow2A=\dfrac{2n}{2n+1}\)
\(\Leftrightarrow A=\dfrac{n}{2n+1}\)
PT
10 tháng 3 2017
\(D= \dfrac{1}{1.3} + \dfrac{1}{3.5}+...+\dfrac{1}{\left(2n-1\right).\left(2n+1\right)}\),
\(2.D = \dfrac{2}{1.3}+ \dfrac{2}{3.5}+...+\dfrac{2}{\left(2n-1\right).\left(2n+1\right)}\)
\(2.D = 1 - \dfrac{1}{3} + \dfrac{1}{3}- \dfrac{1}{5} +\dfrac{1}{5}- \dfrac{1}{7} + ... + \dfrac{1}{\left(2n-1\right)}-\dfrac{1}{\left(2n+1\right)}\)
\(2.D = 1 - \dfrac{1}{\left(2n+1\right)}\)
\(2.D= \dfrac{2n}{\left(2n+1\right)} \)
Vậy \(D = \dfrac{n}{\left(2n+1\right)}\)
PT
10 tháng 3 2017
\(E=\dfrac{1}{1.3.5}+\dfrac{1}{3.5.7}+\dfrac{1}{5.7.9}+...+\dfrac{1}{\left(2n-1\right).\left(2n+1\right).\left(2n+3\right)}\)
\(\Rightarrow4E=4.\dfrac{1}{1.3.5}+\dfrac{1}{3.5.7}+\dfrac{1}{5.7.9}+...+\dfrac{1}{\left(2n-1\right).\left(2n+1\right).\left(2n+3\right)}\)
\(=\dfrac{4}{1.3.5}+\dfrac{4}{3.5.7}+...+\dfrac{4}{\left(2n-1\right).\left(2n+1\right).\left(2n+3\right)}\)
\(=\dfrac{1}{1.3}-\dfrac{1}{3.5}+\dfrac{1}{3.5}-\dfrac{1}{5.7}-...+\dfrac{1}{\left(2n-1\right).\left(2n+1\right)}-\dfrac{1}{\left(2n+1\right).\left(2n+3\right)}\)
\(=\dfrac{1}{1.3}-\dfrac{1}{\left(2n+1\right).\left(2n+3\right)}\)
\(\Rightarrow E=\dfrac{\dfrac{1}{1.3}-\dfrac{1}{\left(2n+1\right).\left(2n+3\right)}}{4}\)
\(=\dfrac{1}{12}-\dfrac{1}{\left(2n+1\right).\left(2n+3\right).4}\)
DB
3
VN
Võ Ngọc Phương
VIP
8 tháng 8 2023
a) \(\dfrac{1}{1\times3}+\dfrac{1}{3\times5}+\dfrac{1}{5\times7}+...+\dfrac{1}{x\times\left(x+3\right)}=\dfrac{99}{200}\)
Ta có: \(\left(1-\dfrac{1}{3}\right)\times\dfrac{1}{2}+\left(\dfrac{1}{3}-\dfrac{1}{5}\right)\times\dfrac{1}{2}+\left(\dfrac{1}{5}-\dfrac{1}{7}\right)\times\dfrac{1}{2}+...+\left(\dfrac{1}{x}-\dfrac{1}{x+3}\right).\dfrac{1}{2}=\dfrac{99}{200}\)
\(\dfrac{1}{2}\times\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{x}-\dfrac{1}{x+3}\right)=\dfrac{99}{200}\)
\(\dfrac{1}{2}\times\left(1-\dfrac{1}{x+3}\right)=\dfrac{99}{200}\)
\(1-\dfrac{1}{x+3}=\dfrac{99}{200}:\dfrac{1}{2}\)
\(1-\dfrac{1}{x+3}=\dfrac{99}{100}\)
\(\dfrac{1}{x+1}=1-\dfrac{99}{100}\)
\(\dfrac{1}{x+1}=\dfrac{1}{100}\)
\(\Rightarrow x+1=100\)
\(x=100-1\)
\(x=99\)
NT
Nguyễn Thị Thương Hoài
Giáo viên
VIP
19 tháng 8 2023
a)\(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{99.101}\)
= \(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{99}-\dfrac{1}{101}\)
= \(1-\dfrac{1}{101}\)
=\(\dfrac{100}{101}\)
NT
Nguyễn Thị Thương Hoài
Giáo viên
VIP
19 tháng 8 2023
\(\dfrac{5}{1.3}+\dfrac{5}{3.5}+\dfrac{5}{5.7}+...+\dfrac{5}{99.101}\)
=\(\dfrac{5}{2}.\left(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{99+101}\right)\)
=\(\dfrac{5}{2}.\left(\dfrac{1}{1}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{99}-\dfrac{1}{101}\right)\)
=\(\dfrac{5}{2}.\left(1-\dfrac{1}{101}\right)\)
= \(\dfrac{5}{2}-\dfrac{100}{101}\)
= \(\dfrac{305}{202}\)
HT
27 tháng 10 2020
sửa đề câu a và câu b nhá , mik nghĩ đề như này :
\(\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\frac{2}{5\cdot7}+\frac{2}{7\cdot9}+...+\frac{2}{213\cdot215}\)
\(=\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{213}-\frac{1}{215}\)
= \(\frac{1}{1}-\frac{1}{215}\)
\(=\frac{214}{215}\)
b, đặt \(A=\frac{1}{1\cdot3}+\frac{1}{3\cdot5}+\frac{1}{5\cdot7}+\frac{1}{7\cdot9}+...+\frac{1}{213\cdot215}\)
\(A\cdot2=\frac{2}{1\cdot3}+\frac{2}{3.5}+\frac{2}{5\cdot7}+\frac{2}{7\cdot9}+...+\frac{2}{213\cdot215}\)
\(A\cdot2=\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{213}-\frac{1}{215}\)
\(A\cdot2=\frac{1}{1}-\frac{1}{215}\)
\(A\cdot2=\frac{214}{215}\)
\(A=\frac{214}{215}:2\)
\(A=\frac{107}{215}\)
LB
7
D
4
S
27 tháng 2 2017
\(\left(\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+\frac{1}{9.11}\right)y=\frac{2}{3}\)
=> \(\frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}\right)y=\frac{2}{3}\)
=> \(\frac{1}{2}\left(1-\frac{1}{11}\right)y=\frac{2}{3}\)
=> \(\frac{1}{2}.\frac{10}{11}y=\frac{2}{3}\)
=> \(\frac{5}{11}y=\frac{2}{3}\)
=>y = \(\frac{2}{3}:\frac{5}{11}\)
=> y = \(\frac{22}{15}\)
3 tháng 4 2021
cho mk cái lời giải thích chỗ nhân 1/2 ý mk ko hiểu mong bn thông cảm
1/1x3 + 1/3x5 + 1/5x7 + ... + 1/(2n+1)x(2n+3) = n+1/2n+3
2/1x3 + 2/3x5 + 2/5x7 + ... + 2/(2n+1)x(2n+3) = 2n+2/2n+3
1 - 1/3 + 1/3 - 1/5 + 1/5 - 1/7 + ... + 1/2n+1 - 1/2n+3 = 2n+2/2n+3
1 - 1/2n+3 = 2n+2/2n+3
Bn nào thông minh thế, ra bài này đố Tây lm đc, ai lm đc mk bái lm sư phụ lun, sửa đề đê
Ủng hộ mk nha ^_-