\(A=1\cdot2\cdot3+2\cdot3\cdot4+\cdots+n\left(n+1\right)\left(n+2\right)\)

\(=2\left(2-1\right)\left(2+1\right)+3\left(3-1\right)\left(3+1\right)+\cdots+\left(n+1\right)\left(n+1-1\right)\left(n+1+1\right)\)

\(=2\left(2^2-1\right)+3\left(3^2-1\right)+\cdots+\left(n+1\right)\left\lbrack\left(n+1\right)^2-1\right\rbrack\)

\(=\left\lbrack2^3+3^3+\cdots+\left(n+1\right)^3\right\rbrack-\left(2+3+\cdots+n+1\right)\)

\(=\left\lbrack1^3+2^3+\cdots+\left(n+1\right)^3\right\rbrack-\left(1+2+\cdots+n+1\right)\)

\(=\left(1+2+\cdots+n+1\right)^2-\left(1+2+\cdots+n+1\right)\)

\(=\left\lbrack\frac{\left(n+1\right)\left(n+2\right)}{2}\right\rbrack^2-\frac{\left(n+1\right)\left(n+2\right)}{2}=\frac{\left(n+1\right)^2\cdot\left(n+2\right)^2-2\left(n+1\right)\left(n+2\right)}{4}\)

\(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{6.7.8}\)

\(=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{6.7}-\frac{1}{7.8}\)

\(=\frac{1}{1.2}-\frac{1}{7.8}\)

\(=\frac{1}{2}-\frac{1}{56}\)

\(=\frac{28}{56}-\frac{1}{56}=\frac{27}{56}\)

Dấu . là nhân nha

\(\frac{2}{1.2.3}=\frac{1}{1.2}-\frac{1}{2.3}\)

\(\frac{2}{2.3.4}=\frac{1}{2.3}-\frac{1}{3.4}\)

.......................................

\(\frac{2}{6.7.8}=\frac{1}{6.7}-\frac{1}{7.8}\)

S= \(\frac{1}{1.2}-\frac{1}{7.8}=\frac{27}{56}\)

A = 1x2 + 2x3 + 3x4 + 4x5 + ...+ 99x100

A x 3 = 1x2x3 + 2x3x3 + 3x4x3 + 4x5x3 + ... + 99x100x3

A x 3 = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) + 4x5x(6-3) + ... + 99x100x(101-98)

A x 3 = 1x2x3 + 2x3x4 - 1x2x3 + 3x4x5 - 2x3x4 + 4x5x6 - 3x4x5 + ... + 99x100x101 - 98x99x100.

A x 3 = 99x100x101

A = 99x100x101 : 3

A = 333300 

Ta có:

\(A=1.2+2.3+3.4+...+99.100\)

\(\Rightarrow3A=1.2.\left(3-0\right)+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+99.100.\left(101-98\right)\)

\(\Rightarrow3A=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100\)

\(\Leftrightarrow3A=99.100.101\Leftrightarrow A=\frac{99.100.101}{3}=333300\)

\(B=1.2.3+2.3.4+4.5.6+...+98.99.100\)

\(\Rightarrow4B=1.2.3.\left(4-0\right)+2.3.4.\left(5-1\right)+4.5.6.\left(7-3\right)+...+98.99.100.\left(101-97\right)\)

\(\Rightarrow4B=1.2.3.4+2.3.4.5-1.2.3.4+4.5.6.7-3.4.5.6+...+98.99.100.101-97.98.99.100\)

\(\Leftrightarrow4B=98.99.100.101\Leftrightarrow B=\frac{98.99.100.101}{4}=24497550\)

2A=\(\frac{2}{1.2.3}\)+\(\frac{2}{2.3.4}\)+...+\(\frac{2}{18.19.20}\)

=1/1.2-1/2.3+1/2.3-1/3.4+...+1/18.19-1/19.20

=1/2-1/19.20

A=1/4-1/19.20.2

vậy A<1/4