n^(n+1)>(n+1)^n的數學歸納法(急)

👨🏻‍🏫 學術台數學

[匿名]

2010-01-17 11:45 pm

如題
證明 n^(n+1)>(n+1)^n 當n>=3時

請使用數學歸納法證明

急需感謝

更新1:

n大於等於3

更新2:

除了樓下第一位說不能用數學歸納法外如果有其他解法的也提出來吧@@ 不過還是以數學歸納法為優先 ^^" 畢竟老師說這用數學歸納法可以導出來

回答 (2)

浮浪貢

2010-01-18 5:19 am

✔ 最佳答案

設 n=k時,k^(k+1) > (k+1)^k………..(1)
(由(k+1)^2>k(k+2)…..得(k+1)/k > (k+2)/(k+1)
得[(k+1)/k]^(k+1)> [(k+2)/(k+1)] ^(k+1)…..(2)
(1)*(2) 得(k+1)^(k+2) > (k+2)^(k+1)

參考: 不等式專書

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myisland8132

2010-01-18 1:24 am

I think it is impossible to use M.I. to prove the required result. Here is my proof :

n^(n+1) > (n+1)^n <=> n > (1+1/n)^n
As we know that a_n=(1+1/n)^n is an increasing sequence which is bounded by e < 3. So n^(n+1) > (n+1)^n when n >=3

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