简便计算:1/2+3/4+7/8+15/16+31/32+63/64+127/128+255/256
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简便计算:1/2+3/4+7/8+15/16+31/32+63/64+127/128+255/256
简便计算:1/2+3/4+7/8+15/16+31/32+63/64+127/128+255/256
简便计算:1/2+3/4+7/8+15/16+31/32+63/64+127/128+255/256
设
a=1/2+3/4+7/8+15/16+31/32+63/64+127/128+255/256
b=1/2+1/4+1/8+1/16+ 1/32+ 1/64+ 1/128+ 1/256
那么a+b=8
b+1/256=1/2+1/4+1/8+1/16+ 1/32+ 1/64+ 1/128+ (1/256+1/256)
=1/2+1/4+1/8+1/16+ 1/32+ 1/64+( 1/128+ 1/128)
=1/2+1/4+1/8+1/16+ 1/32+ (1/64+ 1/64)
=……
=1
所以b=1-1/256
a=8-b=8-(1-1/256)
=7+1/256
原式=1-1/2+1-1/4+1-1/8……+1-1/256
=8-(1/2+1/4+1/8+..+1/256)
=8-255/256
=7又1/265
或 1793/265
=1/2+1/2+1/4+3/4+1/8+7/8+1/16+15/16+1/32+31/32+1/64+63/64+1/128+127/128+1/256
=1+1+1++11+1+1/256
=7又1/256
=1/2+(1/2+1/4)+(3/4+1/8)+(7/8+1/16)+(15/16+1/32)+(1/32+1/64)+(63/64+1/128)+(127/128+1/256)=7又1/256