So sánh A=(2/2^2-1)(1/3^2-1)...(1/100^2-1) và -1/2

Sửa đề:

\(A=\left(\dfrac{1}{2^2}-1\right)\left(\dfrac{1}{3^2}-1\right)\left(\dfrac{1}{4^2}-1\right)....\left(\dfrac{1}{100^2}-1\right)\)

\(A=\left(\dfrac{1}{2^2}-\dfrac{2^2}{2^2}\right)\left(\dfrac{1}{3^2}-\dfrac{3^2}{3^2}\right)\left(\dfrac{1}{4^2}-\dfrac{4^2}{4^2}\right)....\left(\dfrac{1}{100^2}-\dfrac{100^2}{100^2}\right)\)

\(A=\dfrac{\left(1-2^2\right)}{2^2}.\dfrac{\left(1-3^2\right)}{3^2}.\dfrac{\left(1-4^2\right)}{4^2}....\dfrac{\left(1-100^2\right)}{100^2}\)

\(A=\dfrac{\left(1-2\right)\left(1+2\right)}{2^2}.\dfrac{\left(1-3\right)\left(1+3\right)}{3^2}.\dfrac{\left(1-4\right)\left(1+4\right)}{4^2}......\dfrac{\left(1-100\right)\left(1+100\right)}{100^2}\)

\(A=\dfrac{-3}{2^2}.\dfrac{-8}{3^2}.\dfrac{-15}{4^2}....\dfrac{-9999}{100^2}\)

Ta xét từ \(2\) đến \(100\) có: \(\dfrac{\left(100-2\right)}{1}+1=99\)

\(50\) là số lẻ nên tích trên là số âm

Hay \(-A=\dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}.\dfrac{3.5}{4.4}.....\dfrac{99.101}{100.100}\)

\(-A=\dfrac{1.3.2.4.3.5....99.101}{2.2.3.3.4.4.....100.100}\)

\(-A=\dfrac{1.2.3....99}{2.3.4....100}.\dfrac{3.4.5....101}{2.3.4....100}\)

\(-A=\dfrac{1}{100}.\dfrac{101}{2}=\dfrac{101}{200}\)

\(A=-\dfrac{101}{200}< -\dfrac{100}{200}=-\dfrac{1}{2}\)