Question

Tính:

(1/2^2-1).(1/3^2-1). ... .(1/100^2-1)

Answer 1

Giải:

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

\(=\dfrac{-3}{2.2}.\dfrac{-8}{3.3}.....\dfrac{-9999}{100.100}\) 

\(=\dfrac{1.-3.2.-4.....99.-101}{2.2.3.3.....100.100}\) 

\(=\dfrac{\left(1.2.....99\right)}{\left(2.3.....100\right)}.\dfrac{\left(-3.-4.....-101\right)}{\left(2.3.....100\right)}\) 

\(=\dfrac{1}{100}.\dfrac{-101}{2}\) 

\(=\dfrac{-101}{200}\) 

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