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\(M=\left(1-\dfrac{1}{4}\right)\left(1-\dfrac{1}{9}\right)\left(1-\dfrac{1}{16}\right)...\left(1-\dfrac{1}{900}\right)\)
\(=\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)\left(1-\dfrac{1}{4^2}\right)...\left(1-\dfrac{1}{30^2}\right)\)
\(=\left(\dfrac{2^2-1}{2^2}\right)\left(\dfrac{3^2-1}{3^2}\right)\left(\dfrac{4^2-1}{4^2}\right)...\left(\dfrac{30^2-1}{30^2}\right)\)
\(=\left(\dfrac{1.3}{2^2}\right)\left(\dfrac{2.4}{3^2}\right)\left(\dfrac{3.5}{4^2}\right)...\left(\dfrac{29.31}{30^2}\right)\)
\(=\left(\dfrac{1.2.3...29}{2.3.4...30}\right).\left(\dfrac{3.4.5...31}{2.3.4...30}\right)=\dfrac{1}{30}.\dfrac{31}{2}=\dfrac{31}{60}\)
\(1-\dfrac{1}{n^2}=\dfrac{n^2-1}{n^2}=\dfrac{\left(n-1\right)\left(n+1\right)}{n^2}\)
Do đó:
\(M=\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)\left(1-\dfrac{1}{4^2}\right)...\left(1-\dfrac{1}{30^2}\right)\)
\(=\dfrac{\left(2-1\right)\left(2+1\right)}{2^2}.\dfrac{\left(3-1\right)\left(3+1\right)}{3^2}.\dfrac{\left(4-1\right)\left(4+1\right)}{4^2}...\dfrac{\left(30-1\right)\left(30+1\right)}{30^2}\)
\(=\dfrac{1.3}{2^2}.\dfrac{2.4}{3^2}.\dfrac{3.5}{4^2}...\dfrac{29.31}{30^2}=\dfrac{1.2.3...29}{2.3.4...30}.\dfrac{3.4.5...31}{2.3.4...30}\)
\(=\dfrac{1}{30}.\dfrac{31}{2}=\dfrac{31}{60}\)
a) e chỉ cần nhân chúng lại với nhau = cách tách từng cái ra
b)đặt 4/2.5+4/5.8+4/8.11+......+4/62.65 là S
\(.S=\frac{4}{3}\left(\frac{3}{2.5}+\frac{3}{5.8}+...+\frac{3}{62.65}\right)\)
\(S=\frac{4}{3}\left(\frac{1}{2}-\frac{1}{5}+\frac{1}{5}-\frac{1}{8}+...+\frac{1}{62}-\frac{1}{65}\right)\)
\(S=\frac{4}{3}\left(\frac{1}{2}-\frac{1}{65}\right)\)
\(S=\frac{4}{3}\left(\frac{65}{130}-\frac{2}{130}\right)\)
\(S=\frac{4}{3}\left(\frac{63}{130}\right)\)
\(S=\frac{42}{65}\)
Bài 2 : \(\frac{15+a}{29+a}=\frac{3}{5}\)\(\Leftrightarrow\left(15+a\right)5=\left(29+a\right)3\Leftrightarrow75+5a=87+3a\Leftrightarrow5a-3a=87-75\Rightarrow2a=12\Rightarrow a=6\)
vậy a =6
Bài 1:
a) \(\frac{13.23-13}{26.21+26}\)=\(\frac{13.22}{26.22}\)=\(\frac{13}{26}\)=\(\frac{1}{2}\)
b) \(\frac{16.21-16}{4.19+4}\)=\(\frac{16.20}{4.20}\)=\(\frac{16}{4}\)= 4
A=1/2^2+1/3^2+...+1/23^2
=>A<1-1/2+1/2-1/3+...+1/22-1/23
=>A<22/23
a) Ta có: \(\dfrac{7\cdot25}{14\cdot10}\)
\(=\dfrac{7\cdot5\cdot5}{7\cdot2\cdot2\cdot5}\)
\(=\dfrac{5}{4}\)
b) Ta có: \(\dfrac{24\cdot15-14\cdot9}{36\cdot12}\)
\(=\dfrac{9\cdot8\cdot5-14\cdot9}{36\cdot12}\)
\(=\dfrac{9\cdot\left(8\cdot5-14\right)}{9\cdot4\cdot12}\)
\(=\dfrac{40-14}{4\cdot12}\)
\(=\dfrac{13}{24}\)