1⼀4+1⼀8+1⼀16+1⼀32+1⼀64+1⼀128+1⼀512=?

1/4+1/8+1/16+1/32+1/64+1/128+1/512
= 1/512 × (128 + 64 + 32 + 16 + 8 + 4 + 1)
= 253/512

1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512
= 1/512 × (128 + 64 + 32 + 16 + 8 + 4 + 2 + 1)
= 255/512

你看是哪一个？

解：方法一：1/4+1/8+1/16+1/32+1/64+1/128+1/512
=1/(2^2)+1/(2^3)+1/(2^4)+1/(2^5)+1/(2^6)+1/(2^7)+1/(2^8)
=1/4(1-1/(2^7))/(1-1/2) //等比数列求和
=511/1024
方法二：1/4+1/8+1/16+1/32+1/64+1/128+1/512
=1/(2^2)+1/(2^3)+1/(2^4)+1/(2^5)+1/(2^6)+1/(2^7)+1/(2^8)
=(2^10*1/(2^2)+2^10*1/(2^3)+2^10*1/(2^4)+2^10*1/(2^5)+2^10*1/(2^6)+2^10*1/(2^7) +2^10*1/(2^8))/2^(10)
=(2^8+2^7+2^6+2^5+2^4+2^3+2^2)/2^(10)
=511/1024

原式=1/2²+1/2³+...+1/29
等比数列 公比为1/2 剩下的会了吧？

1/4+1/8+1/16+1/32+1/64+1/128+1/512
=1/2-1/512
=255/512