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NM
20 tháng 9 2023
\(a,101^2=101.\left(100+1\right)=10100+101=10201.\\ b,75^2-50.75+25^2\\ =75.\left(75-50\right)+25^2\\ =75.25+25^2\\ =25.\left(75+25\right)\\ =25.100\\ =2500.\)
NM
20 tháng 9 2023
\(c,103.97\\ =\left(100+3\right).97\\ =9700+291\\ =9991\)
C
1
25 tháng 2 2020
\(C=\left(\frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\frac{1}{5\cdot6}+...+\frac{1}{2017\cdot2018}\right)-\)\(\left(\frac{1}{1010}+\frac{1}{1011}+\frac{1}{1012}+...+\frac{1}{2017}\right)\)
Đặt \(A=\frac{1}{1\cdot2}+\frac{1}{3\cdot4}+\frac{1}{5\cdot6}+...+\frac{1}{2017\cdot2018}\)
\(\Rightarrow A=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{2017}-\frac{1}{2018}\)
\(\Rightarrow A=\left(1+\frac{1}{3}+...+\frac{1}{2017}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2018}\right)\)
\(\Rightarrow A=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2017}+\frac{1}{2018}\right)-2\cdot\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2018}\right)\)
\(\Rightarrow A=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2017}+\frac{1}{2018}\right)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2009}\right)\)
\(\Rightarrow A=\frac{1}{1010}+\frac{1}{1011}+\frac{1}{1012}+..+\frac{1}{2017}\)
\(\Rightarrow C=\left(\frac{1}{101}+\frac{1}{1011}+\frac{1}{1012}+...+\frac{1}{2018}\right)-\left(\frac{1}{1010}+\frac{1}{1012}+...+\frac{1}{2017}\right)\)
\(\Rightarrow C=\frac{1}{2018}\)
LT
1
A
23 tháng 11 2016
Đặt \(A=\frac{4}{19.21}+\frac{4}{21.23}+\frac{12}{23.29}+\frac{4}{29.31}\)
\(\Rightarrow\frac{1}{2}A=\frac{2}{19.21}+\frac{2}{21.23}+\frac{6}{23.29}+\frac{2}{29.31}\)
\(\Rightarrow\frac{1}{2}A=\frac{21-19}{19.21}+\frac{23-21}{21.23}+\frac{29-23}{23.29}+\frac{31-29}{29.31}\)
\(\Rightarrow\frac{1}{2}A=\frac{1}{19}-\frac{1}{21}+\frac{1}{21}-\frac{1}{23}+\frac{1}{23}-\frac{1}{29}+\frac{1}{29}-\frac{1}{31}\)
\(\Rightarrow\frac{1}{2}A=\frac{1}{19}-\frac{1}{31}=\frac{12}{589}\)
\(\Rightarrow A=\frac{12}{589}:\frac{1}{2}=\frac{24}{589}\)
DT
17 tháng 2 2022
\(S=\dfrac{1}{2}.\left(\dfrac{2}{\sqrt{1.3}}+\dfrac{2}{\sqrt{3.5}}+.......+\dfrac{2}{\sqrt{29.31}}\right)\)
\(S=\dfrac{1}{2}\left(1-\dfrac{1}{\sqrt{3}}+\dfrac{1}{\sqrt{3}}-\dfrac{1}{\sqrt{5}}+\dfrac{1}{\sqrt{5}}+.....-\dfrac{1}{\sqrt{29}}+\dfrac{1}{\sqrt{29}}-\dfrac{1}{\sqrt{31}}\right)\)
\(S=\dfrac{1}{2}.\left(1-\dfrac{1}{\sqrt{31}}\right)=\dfrac{1}{2}.\left(\dfrac{31-\sqrt{31}}{31}\right)=\dfrac{31-\sqrt{31}}{62}\)
a)
1382-382
=(138-38)2
=1002=10000
b)29x31=899
c)1012=10201