Question

B= (1/2^2-1)(1/3^2-1)(1/4^2-1)...(1/100^2-1)
hãy so sánh B với -1/2

Answer 1

 

Ta thấy: B là tích của 99 số âm

\(\Rightarrow B=\left(1-\dfrac{1}{4}\right)\left(1-\dfrac{1}{9}\right)\left(1-\dfrac{1}{16}\right)...\left(1-\dfrac{1}{100^2}\right)\)

\(=\dfrac{3}{2^2}.\dfrac{8}{3^2}.\dfrac{15}{4^2}...\dfrac{9999}{10^2}\)

\(=\dfrac{1.3}{2^2}.\dfrac{2.4}{3^2}.\dfrac{3.5}{4^2}...\dfrac{99.101}{100^2}\)

\(=\dfrac{1.2.3...98.99}{2.3.4...99.100}.\dfrac{3.4.5...100.101}{2.3.4...99.100}\)

\(=\dfrac{1}{2}.\dfrac{101}{100}\)

\(=\dfrac{101}{200}>\dfrac{1}{2}\)

\(\Rightarrow B< -\dfrac{1}{2}\).

Answer 2

ủa sao từ \(\dfrac{1}{2^2}-1\) lại thành \(1-\dfrac{1}{2^2}\) vậy bạn