\(\text{Đặt }\)\(A=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}\)
\(\Rightarrow2A=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}\)
\(\Rightarrow2A-A=1-\frac{1}{256}\)
\(=>A=\frac{255}{256}\)
1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256=247/256
Ta thấy :
1/2 + 1/4 = 3/4 = 1 - 1/4
1/2 + 1/4 + 1/8 = 7/8 = 1 - 1/8
...............................
1/2 + 1/4 + ... + 1/256 = 1 - 1/256 = 255/256
Vậy 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 = 255/256
ta thấy 1/2=1-1/2, 1/4=1/2-1/4, 1/8=1/4-1/8, ..., 1/256=1/128-1/256
=>1/2+1/4+1/8+1/32+1/64+1/128+1/256
=1- 1/2+1/2-1/4+1/4-1/8+1/8-1/16+1/16-1/32+1/32-1/64+1/64-1/128+1/128-1/256
=1-1/256
=255/256
\(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\frac{1}{128}+\frac{1}{256}\)
\(\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{4}\right)+...+\left(\frac{1}{128}-\frac{1}{256}\right)\)
\(1-\frac{1}{256}=\frac{255}{256}\)
tính nhanh p/s 1+ 5/4 + 5/8 + 5/16 + 5/32 + 5/64
