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Ax3= 1x2x3 + 2x3x3+.....+29x30x3
lấy B= 1x2x3+ 2x3x4+......+29x30x31
B- Ax3= 0 + 1x2x3+ 2x3x4+ 28x29x30= B- 29x30x31. Suy ra Ax3= 29x30x31
![](https://rs.olm.vn/images/avt/0.png?1311)
Ax3= 1x2x3 + 2x3x3+.....+29x30x3 lấy B= 1x2x3+ 2x3x4+......+29x30x31 B- Ax3= 0 + 1x2x3+ 2x3x4+ 28x29x30= B- 29x30x31. Suy ra Ax3= 29x30x31
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\frac{1}{2\times3}+\frac{1}{3\times4}+\frac{1}{4\times5}+....+\frac{1}{29\times30}\)
=\(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+....+\frac{1}{29}-\frac{1}{30}\)
=\(\frac{1}{2}-\frac{1}{30}=\frac{15}{30}-\frac{1}{30}=\frac{14}{30}=\frac{7}{15}\)
\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{19}-\frac{1}{20}\)
\(=\frac{1}{2}-\frac{1}{20}=\frac{10}{20}-\frac{1}{20}=\frac{9}{20}\)
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S = 1.2+2.3+3.4+......+99.100
Gấp S lên 3 lần ta có:
S . 3 = 1.2.3 + 2.3.3 + 3.4.3 + … + 99.100.3
S . 3 = 1.2.3 + 2.3.(4 - 1) + 3.4.( 5 - 2) + … + 99.100. (101 - 98)
S . 3 = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + … + 99.100.101 - 98.99.100
S . 3 = 99.100.101
S = 99.100.101 : 3
S = 33.100.101
S = 333 300
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![](https://rs.olm.vn/images/avt/0.png?1311)
Gọi biểu thức trên là A, ta có :
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