用数学归纳法证明：(n+1)(n+2)(n+3)+.+(n+n)=(2^n)*1*3*.(2n-1)

Question

用数学归纳法证明：(n+1)(n+2)(n+3)+.+(n+n)=(2^n)*1*3*.(2n-1)

Answer 1

n=1时,2=2成立
假设n=k时,(k+1)(k+2)(k+3).(k+k)=(2^k)*1*3*.(2k-1)成立
则当n=k+1时,
(k+2)(k+3).(k+1+k-1)(k+1+k)(k+1+k+1)
=(k+2)(k+3).(k+k)(k+1+k)2(k+1)
=(2^k)*1*3*.(2k-1)*2*(2k+1)
=(2^k+1)*1*3*.(2k-1)(2k+1)
所以：(n+1)(n+2)(n+3).(n+n)=(2^n)*1*3*.(2n-1)
好辛苦 给分吧