(1-1/4)(1-1/9)(1-1/6)*...*(1-1/100)的答案是多少?
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(1-1/4)(1-1/9)(1-1/16)...(1-1/100)
=(1-1/2²)(1-1/3²)(1-1/4²)...(1-1/10²)
=(1+1/2)(1-1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4)...(1+1/10)(1-1/10)
=(3/2)(1/2)(4/3)(2/3)(5/4)(3/4)...(11/10)(9/10)
=[(1/2)(2/3)...(9/10)][(3/2)(4/3)(5/4)...(11/10)]
=[(1×2×...×9)/(2×3×...×10)]×[(3×4×...×11)/(2×3×...×10)]
=(1/10)×(11/2)
=11/20
=(1-1/2²)(1-1/3²)(1-1/4²)...(1-1/10²)
=(1+1/2)(1-1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4)...(1+1/10)(1-1/10)
=(3/2)(1/2)(4/3)(2/3)(5/4)(3/4)...(11/10)(9/10)
=[(1/2)(2/3)...(9/10)][(3/2)(4/3)(5/4)...(11/10)]
=[(1×2×...×9)/(2×3×...×10)]×[(3×4×...×11)/(2×3×...×10)]
=(1/10)×(11/2)
=11/20
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