\(A=\frac{1}{1.101}+\frac{1}{2.102}+\frac{1}{3.103+...}+\frac{1}{10.110}\)

\(A=\frac{1}{100}(\frac{100}{1.101}+\frac{100}{2.102}+\frac{100}{3.103}+...+\frac{100}{10.110})\)

\(A=\frac{1}{100}(\frac{1}{1}-\frac{1}{101}+\frac{1}{2}-\frac{1}{102}+...+\frac{1}{10}-\frac{1}{110})\)

\(A=\frac{1}{100}((\frac{1}{1}+\frac{1}{2}+...+\frac{1}{10})-(\frac{1}{101}+\frac{1}{102}+...+\frac{1}{110}))\)     ok?

\(B=\frac{1}{1.11}+\frac{1}{2.12}+...+\frac{1}{100.110}\)

\(B=\frac{1}{10}(\frac{10}{1.11}+\frac{10}{2.12}+...+\frac{10}{100.110})\)

\(B=\frac{1}{10}(\frac{1}{1}-\frac{1}{11}+\frac{1}{2}-\frac{1}{12}+...+\frac{1}{100}-\frac{1}{110})\)

\(B=\frac{1}{10}((\frac{1}{1}+\frac{1}{2}+...+\frac{1}{100})-(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{110}))\)=\(\frac{1}{10}((\frac{1}{1}+\frac{1}{2}+...+\frac{1}{10})-(\frac{1}{101}+\frac{1}{102}+...+\frac{1}{110}))\)

B=10A 

A.x=10A suy ra x=10

gõ xong mém xỉu. :)

làm biếng gõ quá

Câu hỏi của Huỳnh Ngọc Cẩm Tú - Toán lớp 6 - Học toán với OnlineMath

\(100E\)\(=\frac{100}{1.101}+\frac{100}{2.102}+..........+\frac{100}{10.110}\)

\(=1-\frac{1}{101}+\frac{1}{2}-\frac{1}{102}+........+\frac{1}{10}-\frac{1}{110}\)

\(10F=\frac{10}{1.11}+\frac{10}{2.12}+......+\frac{10}{100.110}\)

\(=1-\frac{1}{11}+\frac{1}{2}-\frac{1}{12}+......+\frac{1}{100}-\frac{1}{110}\)

\(=1+\frac{1}{2}+...+\frac{1}{10}+\frac{1}{11}+....+\frac{1}{100}-\frac{1}{11}-\frac{1}{12}-....-\frac{1}{100}-\frac{1}{101}-...-\frac{1}{110}\)

\(=1+\frac{1}{2}+...+\frac{1}{10}-\frac{1}{101}-\frac{1}{102}-...-\frac{1}{110}\)\(=100E\)

\(\Rightarrow10F=100E\Rightarrow\frac{E}{F}=\frac{1}{10}\)

$100E$$=\genfrac{}{}{0ex}{}{100}{1.101}+\genfrac{}{}{0ex}{}{100}{2.102}+..........+\genfrac{}{}{0ex}{}{100}{10.110}$

$=1-\genfrac{}{}{0ex}{}{1}{101}+\genfrac{}{}{0ex}{}{1}{2}-\genfrac{}{}{0ex}{}{1}{102}+........+\genfrac{}{}{0ex}{}{1}{10}-\genfrac{}{}{0ex}{}{1}{110}$

$10F=\genfrac{}{}{0ex}{}{10}{1.11}+\genfrac{}{}{0ex}{}{10}{2.12}+......+\genfrac{}{}{0ex}{}{10}{100.110}$

$=1-\genfrac{}{}{0ex}{}{1}{11}+\genfrac{}{}{0ex}{}{1}{2}-\genfrac{}{}{0ex}{}{1}{12}+......+\genfrac{}{}{0ex}{}{1}{100}-\genfrac{}{}{0ex}{}{1}{110}$

$=1+\genfrac{}{}{0ex}{}{1}{2}+...+\genfrac{}{}{0ex}{}{1}{10}+\genfrac{}{}{0ex}{}{1}{11}+....+\genfrac{}{}{0ex}{}{1}{100}-\genfrac{}{}{0ex}{}{1}{11}-\genfrac{}{}{0ex}{}{1}{12}-....-\genfrac{}{}{0ex}{}{1}{100}-\genfrac{}{}{0ex}{}{1}{101}-...-\genfrac{}{}{0ex}{}{1}{110}$

$=1+\genfrac{}{}{0ex}{}{1}{2}+...+\genfrac{}{}{0ex}{}{1}{10}-\genfrac{}{}{0ex}{}{1}{101}-\genfrac{}{}{0ex}{}{1}{102}-...-\genfrac{}{}{0ex}{}{1}{110}$$=100E\; lm\; như\; bn\; này\; nha\; bn$

⇒10F=100E⇒EF=110

E = 1/1.101+1/2.102+...+1/10.110

E = 1/100[100/1.101+100/2.102+...+100/10.110]

E = 1/100[1/1-1/101+1/2-1/102+...+1/10-1/110]

E = 1/100[[1/1+1/2+1/3...+1/10]-[1/101+1/102+...+1/110] - xg cái E

F = 1/1.11+1/2.12+...+1/100.110

F = 1/10[10/1.11+10/2.12+...+10/100.110]

F = 1/10[1/1-1/11+1/2-1/12+...+1/100-1/110]

F = 1/10[[1/1+1/2+...+1/100]-[1/11+1/12...+1/110]]

F = 1/10[[1/1+1/2+...+1/10]-[1/101+1/102+...+1/110]

\(\Rightarrow\frac{E}{F}=\frac{\frac{1}{100}\left[\left[\frac{1}{1}+\frac{1}{2}+...+\frac{1}{10}\right]-\left[\frac{1}{101}+\frac{1}{102}+...+\frac{1}{110}\right]\right]}{\frac{1}{10}\left[\left[\frac{1}{1}+\frac{1}{2}+...+\frac{1}{10}\right]-\left[\frac{1}{101}+\frac{1}{102}+...+\frac{1}{110}\right]\right]}=\frac{1}{10}\)

Xỉu... vì đuối sau khi bấm

<br class="Apple-interchange-newline"><div id="inner-editor"></div>⇒EF =1100 [[11 +12 +...+110 ]−[1101 +1102 +...+1110 ]]110 [[11 +12 +...+110 ]−[1101 +1102 +...+1110 ]] =110 

Vì gõ trên Hoc24 khá lâu nên mình gửi hình ảnh cho lẹ

$\left(\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{103.105}\right).\left(x-1\right)=\frac{3}{5}x-\frac{7}{15}\\ \Leftrightarrow \frac{1}{2}.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{103}-\frac{1}{105}\right).\left(x-1\right)=\frac{3}{5}x-\frac{7}{15}\\ \Leftrightarrow \frac{1}{2}.\left(1-\frac{1}{105}\right).\left(x-1\right)=\frac{3}{5}x-\frac{7}{15}\\ \Leftrightarrow \frac{52}{105}.\left(x-1\right)=\frac{3}{5}x-\frac{7}{15}\\ \Leftrightarrow \frac{52}{105}x-\frac{52}{105}=\frac{3}{5}x-\frac{7}{15}\\ \Leftrightarrow x=-\frac{3}{11}$

b) Đặt \(A=\frac{1}{1.101}+\frac{1}{2.102}+\frac{1}{3.103}+...+\frac{1}{10.110}\)

A\(=\frac{1}{100}\left(\frac{100}{1.101}+\frac{100}{2.102}+\frac{1}{3.103}+...+\frac{100}{10.110}\right)\)

A\(=\frac{1}{100}\left(1-\frac{1}{101}+\frac{1}{2}-\frac{1}{102}+...+\frac{1}{10}-\frac{1}{110}\right)\)

A\(=\frac{1}{100}\left[\left(1+\frac{1}{2}+...+\frac{1}{10}\right)-\left(\frac{1}{101}+\frac{1}{102}+...+\frac{1}{110}\right)\right]\)Đặt \(B=\frac{1}{1.11}+\frac{1}{2.12}+...+\frac{10}{100.110}\)

\(B=\frac{1}{10}\left(\frac{10}{1.11}+\frac{10}{2.12}+...+\frac{10}{100.110}\right)\)

\(B=\frac{1}{10}\left(1-\frac{1}{11}+\frac{1}{2}-\frac{1}{12}+...+\frac{1}{100}-\frac{1}{110}\right)\)

\(B=\frac{1}{10}\left[\left(1+\frac{1}{2}+...+\frac{1}{100}\right)-\left(\frac{1}{11}+\frac{1}{12}+...+\frac{1}{110}\right)\right]\)\(=\frac{1}{10}\left[\left(1+\frac{1}{2}+...+\frac{1}{10}\right)-\left(\frac{1}{101}+\frac{1}{102}+...+\frac{1}{110}\right)\right]\)\(B=10A\)

\(A.x=10A\)

\(=>x=10\)

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