Question

用数学归纳法证明(1•22-2•32)+(3•42-4•52)+…+[(2n-1...

用数学归纳法证明(1•22-2•32)+(3•42-4•52)+…+[(2n-1)(2n)2-2n(2n+1)2]=-n(n+1)(4n+3).

Answer 1

证明:当n=1时,左边=-14,右边=-1•2•7=-14,等式成立
假设当n=k时等式成立,
即有(1•22-2•32)+(3•42-4•52)++[(2k-1)(2k)2-2k(2k+1)2]
=-k(k+1)(4k+3)
那么当n=k+1时,
(1•22-2•32)+(3•42-4•52)++[(2k-1)(2k)2-2k(2k+1)2]
+[(2k+1)(2k+2)2-(2k+2)(2k+3)2]
=-k(k+1)(4k+3)-2(k+1)[4k2+12k+9-4k2-6k-2]
=-(k+1)[4k2+3k+2(6k+7)]=-(k+1)[4k2+15k+14]
=-(k+1)(k+2)(4k+7)=-(k+1)[(k+1)+1][4(k+1)+3].
这就是说,当n=k+1时等式也成立.
根据以上论证可知等式对任何n∈N都成立.