MA多伦多大学 MAT 137课业解析

MA多伦多大学 MAT 137课业解析

题意：

完成三道计算题

解析：

第三题： . For which positive integers n ≥ 1 does 2^n > n^2 hold? Prove your claim by induction.

证明：

n>=5

（1）当 n=5 时，2^5=32 > 5^2=25，不等式成立

（2）假设 n=k (k>5)时，2^k > k^2;

则 n = k+1 时，2(k+1)=22k > 2(k2)=(k-1)2-2+(k+1)^2 当k>5时，(k-1)^2-2>0 所以 2(k+1)>(k+1)2 即 n>5 时，假设成立 由数学归纳法可知，V n>=5，2n>n2。

涉及知识点：

数学归纳法，集合

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