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题目:论“对于每一个正整数,如果它是奇数,则对它乘3再加1,如果它是偶数,则对它除以2,如此循环,最终都能够得到1.”结论的成立chuwu8● com
摘要:本文通过双反归纳法实现了对论题的证明
正文:
n为偶数,n/2为偶数,……,一直除2到1;n为偶数,n/2为偶数,一直到n除以2的X次方,为奇数chuwu8● com我们把,n除以2的X次方表示为n,可以等同于n为奇数chuwu8● com(为偶数时,数字一定在减小)
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n为奇数,n×2+n×1+12n+n+1,这个一定为偶数,(2n+n+1)/2n+(n+1)/2,这里又有两种情况,为偶数,为奇数;为偶数就循环①(为偶数时数字一直在减小),一直到n+(n+1)/2为奇数chuwu8● com
因为:n为奇数,有且只有(n+1)/2为偶数1n+(n+1)/2才能为奇数chuwu8● com
n为奇数、n+(n+1)/2为奇数,下面继续:
n+(n+1)/2为奇数,×2+×1+12n+n+1+n+(n+1)/2+1,为偶数,除以
继续两种情况,为偶数,为奇数,为偶数就循环①、②,(反正偶数时数字在减小)
,一直到2n+1+(n+1)/4为奇数chuwu8● com变换为
因为:n为奇数,n+1为偶数,有且仅有(n+1)/4为偶数,n+n+1+(n+1)/4才能为奇数chuwu8● com
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