$(2^0+2^1+······+2^k)*2= (2^1+2^2+······+2^{k+1})$

把p设为$$(2^0+2^1+······+2^n)

那么$2p-p=(2^{k+1}+2^k+······+2^1)-(2^k+2^{k-1} +······+2^0$

去掉括号后在交换一下位置就是$2^{k+1}-2^k+2^k-2^{k-1}+······-2^1+2^1-2^0$

然后消掉相同的最后就是$2^{k+1}-1$

把$2^0$转换为数字$1$

得出的结果就是$2^{k+1}-1$

那么$2^0+2^1+······+2^k=2^{k+1}-1$

太无聊了发的，希望对某些人能有点帮助