Question

1.2+2.3+3.4+.......+99.100

Answer 1

Đặt A = 1.2+2.3+...+99.10

3A = 1.2.3 + 2.3.3 + 3.3.4 + .... + 3.99.100 
Đặt M = 1.2.3 + 2.3.4 + 3.4.5 + .... + 99.100.101 
=> M - 3A = 1.2.3 - 1.2.3 + 2.3.(4-3) + 3.4 ( 5-3) + .... + 99.100 ( 101 -3) 
= 1.2.3 + 2.3.4 + .... + 98.99.100 
=> M -3A = M - 99.100.101 
=> A = 99.100.101/3 = 333300

Answer 2

\(\text{Đặt S= 1.2 + 2.3 + 3.4 + ...+ 99.100}\)

\(\text{ 3S = 1.2.3+2.3.3+3.4.3+...+98.99.3+99.100.3}\)

\(\text{3S= 1.2.3+2.3(4-1)+3.4(5-2)+...+98.99(100-97)+99.100(101-98)}\)

\(\text{3S= 1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...-97.98.99+99.100.101-98.99.100}\)

\(\text{3S = 99.100.101 }\)

\(\text{ S=33.100.101= 333300}\)

Answer 3

Đặt biểu thức trên là A, ta có:

3A = 1 . 2 . 3 + 2 . 3 . 3 + 3 . 3 . 4 + .... + 3 . 99 . 100 

Đặt M = 1 . 2 . 3 + 2 . 3 . 4 + 3 . 4 . 5 + .... + 99 . 100. 101 

=> M - 3A = 1 . 2 . 3  - 1 . 2 . 3 + 2 . 3 . ( 4 - 3 ) + 3 . 4 . ( 5 - 3 ) + .... + 99 . 100 . ( 101 - 3 ) 

= 1 . 2 . 3 + 2 . 3 . 4 + .... + 98 . 99 . 100 

=> M - 3A = M - 99 . 100 . 101 

=> A = 99 . 100 . 101/3 = 333300

Answer 4

\(\text{Đặt S= 1.2 + 2.3 + 3.4 + ...+ 99.100}\)

\(\text{ 3S = 1.2.3+2.3.3+3.4.3+...+98.99.3+99.100.3}\)

\(\text{3S= 1.2.3+2.3(4-1)+3.4(5-2)+...+98.99(100-97)+99.100(101-98)}\)

\(\text{3S= 1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...-97.98.99+99.100.101-98.99.100}\)

\(\text{3S = 99.100.101 }\)

\(\text{ S=33.100.101= 333300}\)