a: Ta có công thức tổng quát:

\(1-\frac{1}{1+2+\cdots+n}\)

\(=1-\frac{1}{\frac{n\left(n+1\right)}{2}}=1-\frac{2}{n\left(n+1\right)}\)

\(=\frac{n\left(n+1\right)-2}{n\left(n+1\right)}=\frac{n^2+n-2}{n\left(n+1\right)}=\frac{\left(n+2\right)\left(n-1\right)}{n\left(n+1\right)}\)

Ta có: \(A=\left(1-\frac{1}{1+2}\right)\left(1-\frac{1}{1+2+3}\right)\cdot\ldots\cdot\left(1-\frac{1}{1+2+\cdots+2022}\right)\)

\(=\frac{\left(2+2\right)\left(2-1\right)}{2\left(2+1\right)}\cdot\frac{\left(3+2\right)\left(3-1\right)}{3\left(3+1\right)}\cdot\ldots\cdot\frac{\left(2022+2\right)\left(2022-1\right)}{2022\left(2022+1\right)}\)

\(=\frac{4\cdot5\cdot\ldots\cdot2024}{3\cdot4\cdot\ldots\cdot2023}\cdot\frac{1\cdot2\cdot\ldots\cdot2021}{2\cdot3\cdot\ldots\cdot2022}=\frac{2024}{3}\cdot\frac{1}{2022}=\frac{1012}{1011\cdot3}=\frac{1012}{3033}\)

b:Sửa đề: \(B=1+\frac12\left(1+2\right)+\frac13\left(1+2+3\right)+\cdots+\frac{1}{100}\left(1+2+\cdots+100\right)\)

\(=1+\frac12\cdot\frac{2\cdot3}{2}+\frac13\cdot\frac{3\cdot4}{2}+\cdots+\frac{1}{100}\cdot\frac{100\cdot101}{2}\)

\(=1+\frac32+\frac42+\cdots+\frac{101}{2}=\frac12\left(2+3+4+\cdots+101\right)\)

\(=\frac12\left(101-2+1\right)\cdot\frac{101+2}{2}=\frac12\cdot100\cdot\frac{101+2}{2}=103\cdot25=2575\)

i giúp em vớiiiiii

\(M=\dfrac{3}{1+2}+\dfrac{3}{1+2+3}+...+\dfrac{3}{1+2+...+2022}\)

\(=\dfrac{3}{\dfrac{2\left(2+1\right)}{2}}+\dfrac{3}{\dfrac{3\left(3+1\right)}{2}}+...+\dfrac{3}{\dfrac{2022\left(2022+1\right)}{2}}\)

\(=\dfrac{6}{2\left(2+1\right)}+\dfrac{6}{3\left(3+1\right)}+...+\dfrac{6}{2022\cdot2023}\)

\(=\dfrac{6}{2\cdot3}+\dfrac{6}{3\cdot4}+...+\dfrac{6}{2022\cdot2023}\)

\(=6\left(\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{2022\cdot2023}\right)\)

\(=6\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2022}-\dfrac{1}{2023}\right)\)

\(=6\cdot\left(\dfrac{1}{2}-\dfrac{1}{2023}\right)=6\cdot\dfrac{2021}{4046}=\dfrac{12126}{4046}< 3\)

mà \(3< \dfrac{10}{3}\)

nên \(M< \dfrac{10}{3}\)

\(\dfrac{3}{5}+\dfrac{4}{7}+\dfrac{2}{5}+\dfrac{1}{7}+\dfrac{2}{7}\\ =\left(\dfrac{3}{5}+\dfrac{2}{5}\right)+\left(\dfrac{4}{7}+\dfrac{1}{7}+\dfrac{2}{7}\right)\\ =\dfrac{5}{5}+\dfrac{7}{7}\\ =1+1\\ =2\)

   3/5 + 4/7 + 2/5 + 1/7 + 2/7

=(3/5 + 2/5) + (4/7 + 1/7 + 2/7)

= 1 + 1 = 2

3/5 và 2/5 có mẫu chung là 5, 2+3 = 5 => 5/5 = 1

4/7, 1/7 và 2/7 có mẫu chung là 7, 4+1+2 = 7 => 7/7 = 1

-Ta có công thức với n∈N* thì:\(1+2+...+n=\dfrac{\left(\dfrac{n-1}{1}+1\right)\left(n+1\right)}{2}=\dfrac{n\left(n+1\right)}{2}\)

\(B=1+\dfrac{1}{2}.\left(1+2\right)+\dfrac{1}{3}.\left(1+2+3\right)+...+\dfrac{1}{2022}.\left(1+2+3+...+2022\right)\)

\(=1+\dfrac{1}{2}.\dfrac{2.3}{2}+\dfrac{1}{3}.\dfrac{3.4}{2}+...+\dfrac{1}{2022}.\dfrac{2022.2023}{2}\)

\(=\dfrac{2}{2}+\dfrac{3}{2}+\dfrac{4}{2}+...+\dfrac{2023}{2}\)

\(=\dfrac{2+3+4+...+2023}{2}=\dfrac{1+2+3+4+...+2022}{2}=\dfrac{\dfrac{2022.2023}{2}}{2}=10222626,5\)

=A*(1/100-1/10^2)

=0

Lời giải:
Đặt $2021=a$ thì:
$A=a^2+(a+1)^2+(a+2)^2+(a+3)^2$
$=4a^2+12a+14=(2a+3)^2+5=4045^2+5$ chia hết cho $25$ nhưng không chia hết cho $5$

Do đó $A$ không là số chính phương 

-----------------------

$9\equiv 1\pmod 4\Rightarrow 9^{100}\equiv 1\pmod 4$

$94^{100}\equiv 0\pmod 4$

$1994^{100}\equiv 0\pmod 4$

$\Rightarrow B\equiv 1+1+0+1\equiv 2\pmod 4$

Một scp không thể chia 4 dư 2 nên $B$ không là scp

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Công thức $1^3+2^3+...+n^3=[\frac{n(n+1)}{2}]^2$ là scp nên $C$ là scp.

mọi người ơi giúp em vs ạ , e đang rất cần 

\(1+2+...+n=\dfrac{\left(\dfrac{n-1}{1}+1\right).\left(n+1\right)}{2}=\dfrac{n\left(n+1\right)}{2}\)

\(M=\dfrac{3}{1+2}+\dfrac{3}{1+2+3}+...+\dfrac{3}{1+2+...+2022}\)

\(=3\left(\dfrac{1}{1+2}+\dfrac{1}{1+2+3}+...+\dfrac{1}{1+2+...+2022}\right)\)

\(=3\left(\dfrac{1}{\dfrac{2.\left(2+1\right)}{2}}+\dfrac{1}{\dfrac{3.\left(3+1\right)}{2}}+...+\dfrac{1}{\dfrac{2022.\left(2022+1\right)}{2}}\right)\)

\(=3\left(\dfrac{2}{2.3}+\dfrac{2}{3.4}+...+\dfrac{2}{2022.2023}\right)\)

\(=3.2.\left(\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{2022.2023}\right)\)

\(=6.\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2022}-\dfrac{1}{2023}\right)\)

\(=6.\left(\dfrac{1}{2}-\dfrac{1}{2023}\right)\)

\(=6.\dfrac{2021}{4046}=3.\dfrac{2021}{2023}=\dfrac{6063}{2023}=\dfrac{18189}{6069}\)

\(\dfrac{10}{3}=\dfrac{20230}{6069}>\dfrac{18189}{6069}=M\)

Bài 1: 

a) \(\dfrac{65}{91}+\dfrac{-33}{55}=\dfrac{5}{7}+\dfrac{-3}{5}=\dfrac{25}{35}+\dfrac{-21}{35}=\dfrac{4}{35}\)

b) \(\dfrac{36}{-84}+\dfrac{100}{450}=\dfrac{-3}{7}+\dfrac{2}{9}=\dfrac{-27}{63}+\dfrac{14}{63}=\dfrac{-13}{63}\)

d) \(\left(\dfrac{1}{2}-\dfrac{1}{3}\right)\left(5-\dfrac{1}{4}\right)=\dfrac{1}{6}\cdot\dfrac{19}{4}=\dfrac{19}{24}\)

e) \(\dfrac{2}{3}-4\left(\dfrac{1}{2}+\dfrac{3}{4}\right)=\dfrac{2}{3}-2-3=\dfrac{2}{3}-\dfrac{15}{3}=\dfrac{-13}{3}\)