Question

CHỨNG MINH RẰNG :

    \(\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}< \frac{1}{3}\)

Answer 1

Ta có: \(\frac{1}{2}-\frac{1}{4}=\frac{2}{4}-\frac{1}{4}=\frac{2-1}{4}=\frac{1}{4}\)

\(\frac{1}{8}-\frac{1}{16}=\frac{2}{16}-\frac{1}{16}=\frac{2-1}{16}=\frac{1}{16}\)

\(\frac{1}{32}-\frac{1}{64}=\frac{2}{64}-\frac{1}{64}=\frac{2-1}{64}=\frac{1}{64}\)

=> \(\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}\)

=\(\frac{1}{4}+\frac{1}{16}+\frac{1}{64}\)

=\(\frac{16}{64}+\frac{4}{64}+\frac{1}{64}=\frac{21}{64}\)

Ta có: \(\frac{21}{64}< \frac{21}{63}=\frac{1}{3}\)

=> \(\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}< \frac{1}{3}\)

Answer 2

Đặt \(A=\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}\)

\(2A=1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\frac{1}{32}\)

\(A+2A=1-\frac{1}{64}\)

\(3A=1-\frac{1}{64}< 1\)

=>A<1/3

=>đpcm

Answer 3

1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64 = 21/64

Nên 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64 < 1/3

Chúc bạn học giỏi

Answer 4

Đặt \(A=\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}\)

\(\Rightarrow2A=1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\frac{1}{32}\)

\(\Rightarrow3A=2A+A=\left(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\frac{1}{32}\right)+\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}\right)\)\(\Rightarrow3A=1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}-\frac{1}{32}+\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\frac{1}{32}-\frac{1}{64}\)

\(\Rightarrow3A=1+\left(\frac{1}{2}-\frac{1}{2}\right)+\left(\frac{1}{4}-\frac{1}{4}\right)+\left(\frac{1}{8}-\frac{1}{8}\right)+\left(\frac{1}{16}-\frac{1}{16}\right)+\left(\frac{1}{32}-\frac{1}{32}\right)-\frac{1}{64}\)

\(\Rightarrow3A=1+0+0+0+0+0-\frac{1}{64}\)

\(\Rightarrow3A=1-\frac{1}{64}< 1\)

\(\Rightarrow A< 1:3\)

\(\Rightarrow A< \frac{1}{3}\left(đpcm\right)\)

Answer 5

1/2-1/4+1/8-1/16+1/32-1/64 là A

thì ta có A= 1/2¹-1/2²+1/2³-1/2⁴+1/2⁵-1/2⁶

<=> 2A=1-1/2¹+1/2²-1/2³+1/2⁴-1/2⁵

<=> A+2A=1-1/2¹+1/2²-1/2³+1/2⁴-1/2⁵+ 1/2¹-1/2²+1/2³-1/2⁴+1/2⁵-1/2⁶

<=> 3A=1-1/2⁶

ta gọi 1/3 là B thì 3B=1 mà 3A=1-1/2⁶ nên sẽ <1

=> 3B>3A=>B>A

vậy1/2-1/4+1/8-1/16+1/32-1/64<1/3

Answer 6

tính phép 1/2 trừ 1/4 cho đến phép 1/64 thì có kết quả chứng minh số đó nhỏ hơn 1/3