Question

Cho B = 1.2 + 2.3 + 3.4 +........+31.32 +32.33. Chứng tỏ rằng B ⋮ 34

Answer 1

\(B=1\cdot2+2\cdot3+3\cdot4+\dots+31\cdot32+32\cdot33\\3B=1\cdot2\cdot3+2\cdot3\cdot3+3\cdot4\cdot3+\dots+31\cdot32\cdot3+32\cdot33\cdot3\\=1\cdot2\cdot3+2\cdot3\cdot(4-1)+3\cdot4\cdot(5-2)+\dots+31\cdot32\cdot(33-30)+32\cdot33\cdot(34-31)\\=1\cdot2\cdot3+2\cdot3\cdot4-1\cdot2\cdot3+3\cdot4\cdot5-2\cdot3\cdot4+\dots+31\cdot32\cdot33-30\cdot31\cdot32+32\cdot33\cdot34-31\cdot32\cdot33\\=32\cdot33\cdot34\\\Rightarrow B=\dfrac{32\cdot33\cdot34}{3}=32\cdot11\cdot34\)

Vì \(32\cdot11\cdot34⋮34\)

nên \(B\vdots34\).

Answer 2

Ta có : B = 1.2 +2.3 +...+31.32 + 32.33  
 3B = 1.2.3 + 2.3.3+...+ 31.32.3 + 32.33.3 
3B = 1.2.3 + 2.3.(4-1) +...+ 31.32.(33-30)+32.33.(34-31)
3B = 1.2.3 + 2.3.4 - 2.3.1 +...+ 31.32.33-30.31.32 + 32.33.34- 31.32.33
3B = 32.33.34                                                                                                                    B = 32.33.34 / 3 = 32.11.34 chia hết cho 34 
Vậy B chia hết cho 34