Question

Cho a/b=1/2+1/3+1/4+...+1/9. Chứng minh a chia hết cho 11

Answer 1

Ta có: \(\dfrac{a}{b}=\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{9}\)

\(\Leftrightarrow\dfrac{1}{1.2}+\dfrac{1}{1.3}+\dfrac{1}{1.4}+....+\dfrac{1}{1.9}\)

Điều cần chứng minh tương ứng với:\(\Leftrightarrow1:\left(1-\dfrac{1}{2}\right)+2:\left(1-\dfrac{1}{3}\right)+3:\left(1-\dfrac{1}{4}\right)+...+8:\left(1-\dfrac{1}{9}\right)⋮11\)

\(\Leftrightarrow\dfrac{1}{2}+\left(2:\dfrac{2}{3}\right)+\left(3:\dfrac{3}{4}\right)+...+\left(8:\dfrac{8}{9}\right)⋮11\)

\(\Leftrightarrow\dfrac{1}{2}+3+4+...+9⋮11\)

\(\Leftrightarrow\dfrac{1}{2}+\left(3+4+...+9\right)=\dfrac{1}{2}+42=\dfrac{85}{2}\)

Answer 2

ko chắc lắm, làm đại đó