1/2.4+1/4.6+.....+1/2n(2n+2)=502/2009 tim n
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Tìm số tự nhiên n thõa mãn
1/2*4+1/4*6+1/6*8+.............+1/2n(2n+2)=502/2009
\(\dfrac{1}{2.4}+\dfrac{1}{4.6}+\dfrac{1}{6.8}+...+\dfrac{1}{2n\left(2n+2\right)}=\dfrac{1009}{4038}\) Tìm n?
Ta có \(\dfrac{1}{2\cdot4}+\dfrac{1}{4\cdot6}+\dfrac{1}{6\cdot8}+...+\dfrac{1}{2n\left(2n+2\right)}=\dfrac{1009}{4038}\)
\(\Leftrightarrow\dfrac{2}{2\cdot4}+\dfrac{2}{4\cdot6}+\dfrac{2}{6\cdot8}+...+\dfrac{2}{2n\left(2n+2\right)}=\dfrac{1009}{2019}\)
\(\Leftrightarrow\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+...+\dfrac{1}{2n}-\dfrac{1}{2n+2}=\dfrac{1009}{2019}\)
\(\Leftrightarrow\dfrac{1}{2}-\dfrac{1}{2n+2}=\dfrac{1009}{2019}\)
\(\Leftrightarrow\dfrac{n}{2n+2}=\dfrac{1009}{2019}\)
\(\Leftrightarrow2019n=1009\left(2n+2\right)\)
\(\Leftrightarrow2019n=2018n+2018\)
\(\Leftrightarrow n=2018\)
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\(\frac{1}{2.4}+\frac{1}{4.6}+\frac{1}{6.8}+...+\frac{1}{2n.\left(2n+2\right)}\)
\(\frac{1}{2.4}+\frac{1}{4.6}+\frac{1}{6.8}+...+\frac{1}{2n.\left(2n+2\right)}\))
\(=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+...+\frac{1}{2n}-\frac{1}{2n+2}\right)\)
\(=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{2n+2}\right)\)
\(=\frac{1}{4}-\frac{1}{2.\left(2n+2\right)}\)
\(=\frac{1}{4}-\frac{1}{4n+4}=\frac{1}{4}-\frac{1}{4.\left(n+1\right)}\)
\(=\frac{n+1}{4.\left(n+1\right)}-\frac{1}{4.\left(n+1\right)}=\frac{n+1-1}{4.\left(n+1\right)}=\frac{n}{4.\left(n+1\right)}\)
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bạn ơi mình ko hiểu chỗ \(\frac{1}{4}-\frac{1}{2.\left(2n+2\right)}\)
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thì là do
\(\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{2n+2}\right)=\frac{1}{2}.\frac{1}{2}-\frac{1}{2}.\frac{1}{2n+2}=\frac{1}{4}-\frac{1}{2.\left(2n+2\right)}\)
:)
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Rút gọn :\(A=\frac{1}{1.3}+\frac{1}{2.4}+\frac{1}{3.5}+\frac{1}{4.6}+...+\frac{1}{\left(2n-1\right)\left(2n+1\right)}+\frac{1}{2n\left(2n+2\right)}\)
tự làm là hạnh phúc của mỗi công dân.
Cho A= 2.4+4.6+6.8+...+2n.(2n+2) với n thuộc N sao Chứng Minh rằng A=4n.(n+1).(n+2)chia cho 3
Tìm số tự nhiên n thỏa mãn \(\frac{1}{2\cdot4}+\frac{1}{4\cdot6}+\frac{1}{6\cdot8}+......+\frac{1}{2n\left(2n+2\right)}=\frac{502}{2009}\)
xét \(VT=\frac{2}{2}\left(\frac{1}{2.4}+\frac{1}{4.6}+......+\frac{1}{2n.\left(2n+2\right)}\right)\) (1)
\(=\frac{1}{2}\left(\frac{2}{2.4}+\frac{2}{4.6}+.......+\frac{2}{2n\left(2n+2\right)}\right)\)
\(=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+.......+\frac{1}{2n}-\frac{1}{2n+2}\right)\)
\(=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{2n+2}\right)=\frac{1}{4}-\frac{1}{2\left(2n+2\right)}\)
\(=\frac{1}{4}-\frac{1}{4n+4}\)
mà theo bài ra (1) = \(\frac{502}{2009}\)
<=>\(\frac{1}{4}-\frac{1}{4n+4}=\frac{502}{2009}\)
<=>\(\frac{1}{4n+4}=\frac{1}{4}-\frac{502}{2009}\)
<=>\(\frac{1}{4n+4}=\frac{1}{8036}\)
<=> 4n+4=8036
<=> 4n=8032
<=> n=2008
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=) \(\frac{1}{2}.\left(\frac{2}{2.4}+\frac{2}{4.6}+...+\frac{2}{2n\left(2n+2\right)}\right)=\frac{502}{2009}\)
=) \(\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{2n}-\frac{1}{2n+2}\right)=\frac{502}{2009}\)
=) \(\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{2n+2}\right)=\frac{502}{2009}\)
=) \(\frac{1}{2}-\frac{1}{2n+2}=\frac{502}{2009}:\frac{1}{2}=\frac{1018}{2009}\)
=) \(\frac{1}{2n+2}=\frac{1}{2}-\frac{1018}{2009}=\frac{-27}{4018}\)
=) \(\frac{-1}{-\left(2n+2\right)}=\frac{-27}{4018}\)
=) \(\frac{-27}{27.-\left(2n+2\right)}=\frac{-27}{4018}\)
=) \(27.-\left(2n+2\right)=4018\)
=) \(-\left(2n+2\right)=4018:27=\frac{4018}{27}\)
=) \(2n+2=\frac{-4018}{27}\)
=) \(2n=\frac{-4018}{27}-2=\frac{-4072}{27}\)
=) \(n=\frac{-4072}{27}:2=\frac{-2036}{27}\)
\(\)
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Ta có:
\(\frac{1}{2}.\left(\frac{2}{2\cdot4}+\frac{2}{4\cdot6}+...+\frac{2}{2n\cdot\left(2n+2\right)}\right)=\frac{502}{2009}\)
\(\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{2n}-\frac{1}{2n+2}\right)=\frac{502}{2009}\)
\(\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{2n+2}\right)=\frac{502}{2009}\)
\(\Rightarrow\frac{1}{2}-\frac{1}{2n+2}=\frac{\frac{502}{2009}}{\frac{1}{2}}=\frac{1004}{2009}\)
\(\frac{1}{2n+2}=\frac{1}{2}-\frac{1004}{2009}=\frac{1}{4018}\)
\(\Rightarrow2n+2=4018\)
\(\Rightarrow n=2013\)
Vậy n= 2013
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Xem thêm câu trả lời
tinh tổng :
a. S = 0.2 + 2.4 + 4.6 +.......+ 98.100
b. S= 0.2+ 2.4 + 4.6 + .........+ 2n. ( 2n +2 )
Tinh:
a/S=0.2+2.4+4.6+......+98.100
b/ A= 0.2+2.4+4.6+......+ 2n(2n+2)
Tính các tổng sau:
1.100+2.99+3.98+...+98.3+99.2+100.19+99+999+....+999...9991.2+2.3+3.4+...+n(n+1)2.4+4.6+6.8+....+2n(2n+2)1.3+2.4+3.5+...+n(n+2)1.2.3+2.3.4+3.4.5+....+n(n+1).(n+2)12+22+32+...+n2