Question

Cho A = \(\left(1+\dfrac{1}{2}\right)\left(1+\dfrac{1}{3}\right)\left(1+\dfrac{1}{4}\right).....\left(1+\dfrac{1}{2022}\right)\) . Chứng tỏ rằng A > 1000
 

Answer 1

\(A=\left(1+\dfrac{1}{2}\right)\left(1+\dfrac{1}{3}\right)\left(1+\dfrac{1}{4}\right)....\left(1+\dfrac{1}{2022}\right)\)

\(A=\dfrac{2+1}{2}\cdot\dfrac{3+1}{3}\cdot\dfrac{4+1}{4}\cdot...\cdot\dfrac{2022+1}{2022}\)

\(A=\dfrac{3}{2}\cdot\dfrac{4}{3}\cdot\dfrac{5}{4}\cdot...\cdot\dfrac{2023}{2022}\)

\(A=\dfrac{3\cdot4\cdot5\cdot...\cdot2023}{2\cdot3\cdot4\cdot...\cdot2022}\)

\(A=\dfrac{2023}{2}>\dfrac{2000}{2}\)

\(A>1000\)