用数学归纳法证明1-3/2+5/4-7/8+...+(-1)^(n-1)*2n-1/2^(n-1)=2^n+(-1)^n-1*(6n+1)/9*2^n-1

Question

用数学归纳法证明1-3/2+5/4-7/8+...+(-1)^(n-1)*2n-1/2^(n-1)=2^n+(-1)^n-1*(6n+1)/9*2^n-1

Answer 1

当n = 1时,等式显然成立；
假设：当n = k时,有
1-3/2+5/4-7/8+...+(-1)^(k-1)*2k-1/2^(k-1)=2^k+(-1)^k-1*(6k+1)/9*2^k-1
则对于：n = k+1时,
1-3/2+5/4-7/8+...+(-1)^(k-1)*2k-1/2^(k-1) + (-1)^((k+1)-1)*2(k+1)-1/2^((k+1)-1)=2^k+(-1)^k-1*(6k+1)/9*2^k-1 + (-1)^((k+1)-1)*2(k+1)-1/2^((k+1)-1)
等号右边整理得：2^(k+1)+(-1)^(k+1)-1*(6(k+1)+1)/9*2^(k+1)-1
则假设成立
得结论成立