Question

(1-1/2²)(1-1/3²)(1-1/4²)……(1-1/2012²)(1-1/2013²)等于几，要过程

Answer 1

你好~~很高兴为你提供详细解答~~
(1-1/2²)(1-1/3²)(1-1/4²)……(1-1/2012²)(1-1/2013²)
=[(2²-1)/2²][(3²-1)/3²][(4²-1)/4²]……[(2012²-1)/2012²][(2013²-1)/2013²]
=(2²-1)(3²-1)(4²-1)……(2012²-1)(2013²-1)
÷
(2²×3²×4²……2012²×2013²)
=(2+1)(2-1)(3+1)(3-1)(4+1)(4-1)……(2012+1)(2012-1)(2013+1)(2013-1)
÷
(2013!)²
=[(2-1)(3-1)(4-1)……(2012-1)(2013-1)]
×
[(2+1)(3+1)(4+1)……(2012+1)(2013+1)]
÷(2013!)²
=(1×2×3……2011×2012)
×
(3×4×5……2013×2014)
÷
(2013!)²
=(2012!
×
2014!/2)
÷
(2013!)²
=(2012!
×
1007
×
2013!)
÷
(2013!
×
2013
×
2012!
)
=1007/2013
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