Question

Tìm
A=1.2+2.3+3.4+...+30.31
B=1+(1+2)+(1+2+3)+...+(1+2+...+20)

Answer 1

Trả lời :

A = 1 . 2 + 2 . 3 + 3 . 4 + ... + 30 . 31

=> 3A = 1 . 2 . 3 + 2 . 3 . 3 + 3 . 4 . 3 + ... + 30 . 31 . 3

=> 3A = 1 . 2 . 3 + 2 . 3 . (4 - 1) + 3 . 4 . (5 - 2) + ... + 30 . 31 . (32 - 29)

=> 3A = 1 . 2 . 3 + 2 . 3 . 4 - 1 . 2 . 3 + 3 . 4 . 5 - 2 . 3 . 4 + ... + 30 . 31 . 32 - 29 . 30 . 31

=> 3A = 30 . 31 . 32

=> 3A = 29760

=> A = 9920

Answer 2

a) \(A=1.2+2.3+3.4+.........+30.31\)

\(\Rightarrow3A=1.2.3+2.3.3+3.4.3+........+30.31.3\)

\(=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+.....+30.31.\left(32-29\right)\)

\(=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+......+30.31.32-29.30.31\)

\(=30.31.32\)

\(\Rightarrow A=\frac{30.31.32}{3}=9920\)

b) \(B=1+\left(1+2\right)+\left(1+2+3\right)+........+\left(1+2+...........+20\right)\)

\(=\frac{1.2}{2}+\frac{2.3}{2}+\frac{3.4}{2}+.......+\frac{20.21}{2}\)

\(=\frac{1.2+2.3+3.4+.......+20.21}{2}\)

Làm tương tự như phần a ta được: 

\(1.2+2.3+3.4+.......+20.21=\frac{20.21.22}{3}=3080\)

\(\Rightarrow B=\frac{3080}{2}=1540\)