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DS
Đinh Sơn Tùng
CTVHS
VIP
21 tháng 11 2023
Để tính tổng 11009×2016+11010×2015+…+12015×1010+11016×10091009×20161+1010×20151+…+2015×10101+1016×10091, ta có thể sử dụng một số kỹ thuật trong toán học. Trong trường hợp này, ta sẽ sử dụng tích phân.
Gọi �S là tổng cần tính, ta có thể viết nó dưới dạng tổng tỉ lệ:
�=11009×2016+11010×2015+…+12015×1010+11016×1009S=1009×20161+1010×20151+…+2015×10101+
S
8 tháng 6 2017
Sửa đề: Cho \(V=\frac{1}{1.2}+\frac{1}{3.4}+...+\frac{1}{2015.2016}\)và \(Y=\frac{1}{1009}+\frac{1}{1010}+...+\frac{1}{2016}\). Tính \(\frac{V}{Y}\)
\(V=\frac{1}{1.2}+\frac{1}{3.4}+...+\frac{1}{2015.2016}\)
\(=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2015}-\frac{1}{2016}\)
\(=\left(1+\frac{1}{3}+...+\frac{1}{2015}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2016}\right)\)
\(=\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2016}\right)-2\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2016}\right)\)
\(=\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2016}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{1008}\right)\)
\(=\frac{1}{1009}+\frac{1}{1010}+...+\frac{1}{2016}\)
=> \(\frac{V}{Y}=\frac{\frac{1}{1009}+\frac{1}{1010}+...+\frac{1}{2016}}{\frac{1}{1009}+\frac{1}{1010}+...+\frac{1}{2016}}=1\)
8 tháng 6 2017
V = \(\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+...+\frac{1}{2015.2016}\)
V = \(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{2015}-\frac{1}{2016}\)
V = \(\left(1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{2015}\right)-\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{2016}\right)\)
V = \(\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+...+\frac{1}{2015}+\frac{1}{2016}\right)-2.\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+...+\frac{1}{2016}\right)\)
V = \(\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2015}+\frac{1}{2016}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{1008}\right)\)
V = \(\frac{1}{1009}+\frac{1}{1010}+...+\frac{1}{2016}\)
Vậy V : Y = \(\frac{\frac{1}{1009}+\frac{1}{1010}+...+\frac{1}{2016}}{\frac{1}{1008}+\frac{1}{1009}+...+\frac{1}{2016}}\)
( Mình nghĩ Y = 1/1009 + 1/1010 + ... + 1/2016 / Nếu Y như mình nói thì V : Y = 1 )
NT
1
DM
\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+........+\frac{2}{x\cdot\left[x+1\right]}=\frac{1008}{1009}\)
1
5 tháng 8 2017
\(\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\left(x+1\right)}=\frac{1008}{1009}\)
\(\Leftrightarrow2\left(\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{1008}{1009}\)
\(\Leftrightarrow2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x.\left(x+1\right)}\right)=\frac{1008}{1009}\)
\(\Leftrightarrow2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{1008}{1009}\)
\(\Leftrightarrow2\left(\frac{1}{2}-\frac{1}{x+1}\right)=\frac{1008}{1009}\)
\(\Leftrightarrow1-\frac{2}{x+1}=\frac{1008}{1009}\)
\(\Leftrightarrow\frac{-2}{x-1}=\frac{1008}{1009}-1\)
\(\Leftrightarrow\frac{-2}{x+1}=\frac{-1}{1009}\)
\(\Leftrightarrow-1.\left(x+1\right)=-2.1009\)
\(\Leftrightarrow-x-1=-2018\)
\(\Leftrightarrow-x=-2018+1=-2017\)
\(\Leftrightarrow x=2017\)
Vậy x=2017
5 tháng 5 2016
Xét số chia: 1-\(\frac{1}{2}\) + \(\frac{1}{3}\) - \(\frac{1}{4}\) +...+\(\frac{1}{2015}\) - \(\frac{1}{2016}\)
= (1+\(\frac{1}{2}\) + \(\frac{1}{3}\) + \(\frac{1}{4}\) +...+\(\frac{1}{2015}\) + \(\frac{1}{2016}\)) - 2.(\(\frac{1}{2}\) + \(\frac{1}{4}\) + ... + \(\frac{1}{2016}\))
= (1+\(\frac{1}{2}\) + \(\frac{1}{3}\) + \(\frac{1}{4}\) +...+\(\frac{1}{2015}\) + \(\frac{1}{2016}\)) - (1+\(\frac{1}{2}\) + \(\frac{1}{3}\) + \(\frac{1}{4}\) +...+\(\frac{1}{1007}\) + \(\frac{1}{1008}\))
=\(\frac{1}{1009}\) + \(\frac{1}{1010}\) + ... + \(\frac{1}{2015}\)+ \(\frac{1}{2016}\) => A=1
DT
1
O
8 tháng 4 2018
\(CMR:\dfrac{1}{2}+\dfrac{1}{3}.4+\dfrac{1}{5}.6+...+\dfrac{1}{2015}.2016=\dfrac{1}{1009}+\dfrac{1}{1010}+...+\dfrac{1}{2016}\)
\(=\dfrac{1}{1.2}+\dfrac{1}{3.4}+...+\dfrac{1}{2015.2016}\)
\(=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2015}-\dfrac{1}{2016}\)
\(=1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{2015}+\dfrac{1}{2016}-2\left(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{2016}\right)\)
\(=\dfrac{1}{1009}+\dfrac{1}{1010}+...+\dfrac{1}{2016}\left(đpcm\right)\).
13 tháng 3 2018
\(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2013}-\frac{1}{2014}\)
\(=\left(1+\frac{1}{3}+...+\frac{1}{2013}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2014}\right)\)
\(=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2013}+\frac{1}{2014}-2\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2014}\right)\)
\(=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2013}+\frac{1}{2014}-1-\frac{1}{2}-...-\frac{1}{1007}\)
\(=\frac{1}{1008}+\frac{1}{1009}+\frac{1}{1010}+...+\frac{1}{2014}\) (đpcm)
X = 2019
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