Đặt bt bằng A
Ta có 2A= 2(2/3 + 2/6 + 2/12 +2/24 + 2/48 + 2/96 + 2/192)
2A= 4/3 +2/3 + 2/6 + 2/12 + 2/24 + 2/48 + 2/96
A= 2A-A= (4/3 +2/3 + 2/6 + 2/12 + 2/24 + 2/48 + 2/96) - (2/3 + 2/6 + 2/12 +2/24 + 2/48 + 2/96 2/192)
A=4/3 +2/3 + 2/6 + 2/12 + 2/24 + 2/48 + 2/96 - 2/3 - 2/6 - 2/12 - 2/24 - 2/48 - 2/96 - 2/192
A=(2/3 - 2/3) + (2/6 - 2/6) + ( 2/12 - 2/12) + (2/24 - 2/24) + (2/48 - 2/48) + ( 2/96 - 2/96) + (4/3 - 2/192)
A=0+0+0+0+0+0+ (256/192 - 2/192)
A=254/192
A=127/96(rút gọn phân số)
\(\frac{2}{3}+\frac{2}{6}+\frac{2}{12}+...+\frac{2}{192}\)
\(=2\times\left(\frac{1}{3}+\frac{1}{6}+\frac{1}{12}+...+\frac{1}{192}\right)\)
\(=2\times\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{6}+\frac{1}{6}-\frac{1}{12}+...+\frac{1}{96}-\frac{1}{192}\right)\)
\(=2\times\left(1-\frac{1}{192}\right)\)
\(=2\times\frac{191}{192}\)
\(=\frac{382}{192}=\frac{191}{96}\)
\(\frac{2}{3}+\frac{2}{6}+\frac{2}{12}+\frac{2}{24}+\frac{2}{48}+\frac{2}{96}+\frac{2}{192}\)
\(=2.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{6}+...+\frac{1}{96}-\frac{1}{192}\right)\)
\(=2.\left(1-\frac{1}{192}\right)\)
\(=2.\frac{191}{192}\)
\(=\frac{382}{192}=\frac{191}{96}\)
2/3 + 2/6 + 2/12 + 2/24 + 2/48 + 2/96 + 2/192
= 2/3 x (1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64)
Xét phần trong ngoặc: (1 +1/2 +1/4+1/8+1/16+1/32+1/64)
1/2 + 1/4 = 1 - 1/4
1/2 + 1/4 + 1/8 = 1 - 1/8
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 = 1 - 1/64 = 63/64
Suy ra 1 + 63/64 = 127/64
2/3 + 2/6 + 2/12 + 2/24 + 2/48 + 2/96 + 2/192
= 2/3 x 127/64 = 127/96
=2x(\(\frac{2}{3}-\frac{1}{3}\)+\(\frac{1}{3}-\frac{1}{6}\)+...+ \(\frac{1}{96}-\frac{1}{192}\))=
=2x( \(\frac{2}{3}-\frac{1}{192}\)) = \(\frac{127}{96}\)
