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Câu 9 trang 135 SGK Đại số và Giải tích 11 Nâng cao

Biểu diễn các số thập phân
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Biểu diễn các số thập phân vô hạn tuần hoàn sau dưới dạng phân số :

LG a

 \(0,444…\)

Phương pháp giải:

Sử dụng công thức tính tổng cấp số nhân lùi vô hạn \(S = \frac{{{u_1}}}{{1 - q}}\)

Lời giải chi tiết:

 Ta có:

\(\eqalign{
& 0,444... = 0,4 + 0,04 + 0,004 + ... \cr 
& = {4 \over {10}} + {4 \over {{{10}^2}}} + {4 \over {{{10}^3}}} + ... \cr 
& = 4\left( {{1 \over {10}} + {1 \over {{{10}^2}}} + ...} \right) \cr 
🐼& = 4.{{{1 \over {10}}} \over {1 - {1 \over {10}}}} = {4 \over 9} \cr} \)

LG b

\(0,2121…\)

Lời giải chi tiết:

\(\eqalign{
& 0,2121... = 0,21 + 0,0021 + ... \cr 
& = {{21} \over {{{10}^2}}} + {{21} \over {{{10}^4}}} + ... \cr &= 21\left( {{1 \over {{{10}^2}}} + {1 \over {{{10}^4}}} + ...} \right) \cr 
🍌& = 21.{{{1 \over {{{10}^2}}}} \over {1 - {1 \over {{{10}^2}}}}} = {{21} \over {99}} = {7 \over {33}} \cr} \) .

LG c

\(0,32111…\)

Lời giải chi tiết:

\(\eqalign{
& 0,32111...\cr & = {{32} \over {100}} + {1 \over {1000}} + {1 \over {10000}}+ ... \cr 
𝐆&  = \frac{{32}}{{100}} + \frac{1}{{1000}}\left( {1 + \frac{1}{{10}} + \frac{1}{{{{10}^2}}} + ...} \right)\cr &= {{32} \over {100}} + {1 \over {1000}}.{1 \over {1 - {1 \over {10}}}}\cr & = {{32} \over {100}} + {1 \over {900}} = {{289} \over {900}} \cr} \)

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