230问答网 > 巧算：（1-1⼀2^2)(1-1⼀3^2)(1-1⼀4^2)...(1-1⼀2008^2)

巧算：（1-1⼀2^2)(1-1⼀3^2)(1-1⼀4^2)...(1-1⼀2008^2)

2024-12-21 04:09:02

推荐回答（2个）

回答1：

（1-1/2^2)(1-1/3^2)(1-1/4^2)...(1-1/2008^2)
=(2^2-1)/2^2 ×（3^2-1)/3^2 × (4^2-1)/4^2...(2008^2-1)/2008^2
=[(2+1)(2-1)(3+1)(4-1)(4+1)(4-1)...(2008+1)(2008-1)]/(2^2 × 3^2 × 4^2...2008^2)
=(3×1×4×2×5×3×6×4×7×5...2009×2007)/(2^2 × 3^2 × 4^2...2008^2)
=(3×4×5×6...2009×1×2×3...2007)/(2^2 × 3^2 × 4^2...2008^2)
=(1×2×3^2×4^2×5^2×...2007^2×2008×2009)/(2^2 × 3^2 × 4^2...2008^2)
=(1×2×2008×2009)/(2^2×2008^2)=2009/(2×2008)=2009/5016

回答2：

1-1/2^2)(1-1/3^2)(1-1/4^2)...(1-1/2008^2)
=(2^2-1)/2^2 ×（3^2-1)/3^2 × (4^2-1)/4^2...(2008^2-1)/2008^2
=[(2+1)(2-1)(3+1)(4-1)(4+1)(4-1)...(2008+1)(2008-1)]/(2^2 × 3^2 × 4^2...2008^2)
=(3×1×4×2×5×3×6×4×7×5...2009×2007)/(2^2 × 3^2 × 4^2...2008^2)
=(3×4×5×6...2009×1×2×3...2007)/(2^2 × 3^2 × 4^2...2008^2)
=(1×2×3^2×4^2×5^2×...2007^2×2008×2009)/(2^2 × 3^2 × 4^2...2008^2)
=(1×2×2008×2009)/(2^2×2008^2)=2009/(2×2008)=2009/5016