题目

设n∈N*且n＞1，求证：(1)lgnlg(n+2)＜lg2(n+1)；(2)logn+1n＜logn+2(n+1). 答案：证明：(1)∵n＞1，∴lgn＞0，lg(n+2)＞0.∴lgn·lg(n+2)≤［］2=lg2(n2+2n)＜lg2(n2+2n+1)=lg2(n+1).∴lgnlg(n+2)＜lg2(n+1).(2)由(1)知lgnlg(n+2)＜lg(n+1)lg(n+1)，∴，即logn+1n＜logn+2(n+1).

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