Question

Mấy chế zô giải giùm tui câu này:

Cho \(A=\left(\dfrac{1}{2^2}-1\right)\left(\dfrac{1}{3^2}-1\right)\left(\dfrac{1}{4^2}-1\right).....\left(\dfrac{1}{2013^2}-1\right)\left(\dfrac{1}{2014^2}-1\right)\) và \(B=-\dfrac{1}{2}.\)

Hãy so sánh A và B.

Answer 1

\(A=\left(\dfrac{1}{2^2}-1\right)\left(\dfrac{1}{3^2}-1\right).......\cdot\left(\dfrac{1}{2014^2}-1\right)\)

\(=\left(\dfrac{1}{4}-1\right)\left(\dfrac{1}{9}-1\right)..........\left(\dfrac{1}{4056196}-1\right)\)

\(=\dfrac{-3}{4}.\dfrac{-8}{9}..............\dfrac{-4056195}{4056196}\)

\(=\dfrac{\left(-1\right).3}{2^2}.\dfrac{\left(-2\right).4}{3^2}...........\dfrac{\left(-2013\right).2015}{2014^2}\)

\(=\dfrac{\left(-1\right).\left(-2\right)..........\left(-2013\right)}{2.3......2014}.\dfrac{3.4......2015}{2.3.....2014}\)

\(=\dfrac{-1}{2014}.\dfrac{2015}{2}\)

\(=\dfrac{-2015}{4028}< \dfrac{-1}{2}\)

\(\Leftrightarrow A< B\)