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A=1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64
A= ( 1/2 - 1/4 ) + ( 1/8 - 1/16 ) + ( 1/32 - 1/64 )
A= 1/4 + 1/16 + 1/64
A = 16/64 + 4/64 + 1/64
A = 16+4+1/64
A= 21/64
Ta có : 1/3 = 21/63 mà 21/64 < 21/63 => 21/64 < 1/3 => 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/ 64 < 1/3
Vậy 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/ 64 < 1/3
\(A=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}\)
\(\rightarrow A=\frac{3}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}\)
\(\rightarrow A=\frac{7}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}\)
\(\rightarrow A=\frac{15}{16}+\frac{1}{32}+\frac{1}{64}\)
\(\rightarrow A=\frac{31}{32}+\frac{1}{64}\)
\(\rightarrow A=\frac{63}{64}\)
\(A=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}\Rightarrow64A=32+16+8+4+2+1\Rightarrow64A=63\Rightarrow A=\frac{63}{64}\)
đúng rồi đó Trương Quang Hải ( đừng tik cho Trương Quang Hải)
=(1/2 _ 1/4) + (1/8 + 1/16 ) + ( 1/32 - 1/64 )
= 1/4 +1/16 + 1/64
= 16 + 4 + 1/ 64
= 21/64 < 21/63
= 1/3
=> 1/2 -1/4 + 1/8 - 1/16 + 1/32 - 1/ 64 < 1/3
Chúc bạn làm bài tốt =))
A=1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64
A= ( 1/2 - 1/4 ) + ( 1/8 - 1/16 ) + ( 1/32 - 1/64 )
A= 1/4 + 1/16 + 1/64
A = 16/64 + 4/64 + 1/64
A = 16+4+1/64
A= 21/64
Ta có : 1/3 = 21/63 mà 21/64 < 21/63 => 21/64 < 1/3 => 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/ 64 < 1/3
Vậy 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/ 64 < 1/3 ( đã chứng minh được )
Đặt A= \(\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}\)
=\(\frac{1}{2^1}-\frac{1}{2^2}+\frac{1}{2^3}-\frac{1}{2^4}+\frac{1}{2^5}-\frac{1}{2^6}\)
=> 2A= \(1-\frac{1}{2^1}+\frac{1}{2^2}-\frac{1}{2^3}+\frac{1}{2^4}-\frac{1}{2^5}\)
Ta có:
2A+A=\(\left(1-\frac{1}{2^1}+\frac{1}{2^2}-\frac{1}{2^3}+\frac{1}{2^4}-\frac{1}{2^5}\right)+\left(\frac{1}{2^1}-\frac{1}{2^2}+\frac{1}{2^3}-\frac{1}{2^4}+\frac{1}{2^5}-\frac{1}{2^6}\right)\)
=> 3A=\(1-\frac{1}{2^1}+\frac{1}{2^2}-\frac{1}{2^3}+\frac{1}{2^4}-\frac{1}{2^5}+\frac{1}{2^1}-\frac{1}{2^2}+\frac{1}{2^3}-\frac{1}{2^4}+\frac{1}{2^5}-\frac{1}{2^6}\)
=\(1-\left(\frac{1}{2^1}-\frac{1}{2^1}\right)+\left(\frac{1}{2^2}-\frac{1}{2^2}\right)-\left(\frac{1}{2^3}-\frac{1}{2^3}\right)+\left(\frac{1}{2^4}-\frac{1}{2^4}\right)-\left(\frac{1}{2^5}-\frac{1}{2^5}\right)-\frac{1}{2^6}\)
= \(1-\frac{1}{2^6}\)
=> A= 3A:3= \(\left(1-\frac{1}{2^6}\right):3\)=\(\frac{1}{3}-\frac{1}{2^6}:3\)<\(\frac{1}{3}\)