MA多倫多大學 MAT 137課業解析

MA多倫多大學 MAT 137課業解析

題意：

完成三道計算題

解析：

第三題： . For which positive integers n ≥ 1 does 2^n > n^2 hold? Prove your claim by induction.

證實：

n>=5

（1）當 n=5 時，2^5=32 > 5^2=25，不等式成立

（2）假設 n=k (k>5)時，2^k > k^2;

則 n = k+1 時，2(k+1)=22k > 2(k2)=(k-1)2-2+(k+1)^2 當k>5時，(k-1)^2-2>0 因此 2(k+1)>(k+1)2 即 n>5 時，假設成立 由數學概括法可知，V n>=5，2n>n2。

涉及知識點：

數學概括法，集合

更多可加V討論

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