用数学归纳法证明命题: (n+1)×(n+2)×…×(n+n)=2n×1×3×…×(2n-1)
问题描述:
用数学归纳法证明命题:
(n+1)×(n+2)×…×(n+n)=2n×1×3×…×(2n-1)
答
证明:①当n=1时,左边=2,右边=21×1,等式成立;
②假设当n=k时,等式成立,
即(k+1)×(k+2)×…×(k+k)=2k×1×3×…×(2k-1)
则当n=k+1时,
左边=(k+2)×(k+3)×…×(k+k)×(k+k+1)×(k+1+k+1)
=2k×1×3×…×(2k-1)×(k+k+1)×(k+1+k+1)
=2k×1×3×…×(2k-1)×[2(k+1)-1]×(k+1)×2
=2k+1×1×3×…×(2k-1)×[2(k+1)-1]
即n=k+1时,等式也成立.
所以(n+1)×(n+2)×…×(n+n)=2n×1×3×…×(2n-1)对任意正整数都成立.