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\(\left(1-\frac{1}{3}\right)+\left(1-\frac{1}{15}\right)+...+\left(1-\frac{1}{9999}\right)\)
= \(\left(1-\frac{1}{1.3}\right)+\left(1-\frac{1}{3.5}\right)+...+\left(1-\frac{1}{99.101}\right)\)(50 cặp)
= \(\left(1+1+1+...+1\right)-\left(\frac{1}{1.3}+\frac{1}{3.5}+...+\frac{1}{99.101}\right)\)(50 số hạng 1)
= \(1.50-\frac{1}{2}.\left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{99.101}\right)\)
= \(50-\frac{1}{2}.\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{101}\right)\)
= \(50-\frac{1}{2}.\left(1-\frac{1}{101}\right)\)
= \(50-\frac{1}{2}.\frac{100}{101}\)
= \(50-\frac{50}{101}\)
= \(\frac{5000}{101}\)
2/3+2/15+2/35+2/63+...+2/9999
=2/1.3+2/3.5+2/5.7+...+2/99x101
=1-1/3+1/3-1/5+...+1/99-1/101
=1-1/101=100/101
\(=2\times\left(\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+...+\frac{1}{9999}\right)\)
\(=2\times\left(\frac{1}{1\times3}+\frac{1}{3\times5}+\frac{1}{5\times7}+...+\frac{1}{99\times101}\right)\)
\(=2\times\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{99}-\frac{1}{101}\right)\)
\(=2\times\left(\frac{1}{1}-\frac{1}{101}\right)\)
\(=2\times\frac{100}{101}\)
\(=\frac{200}{101}\)
3/4.8/9.15/16...9999/10000
=\(\dfrac{1.3}{2.2}\).\(\dfrac{2.4}{3.3}\)...\(\dfrac{99.101}{100.100}\)
=\(\dfrac{1.2...99}{2.3.100}\).\(\dfrac{3.4...101}{2.3.100}\)
=\(\dfrac{1}{100}\).\(\dfrac{101}{2}\)
=\(\dfrac{101}{200}\)
\(\frac{3}{4}+\frac{8}{9}+\frac{15}{16}\)+...+\(\frac{9999}{10000}\)
= (1-\(\frac{1}{4}\)) +(1-\(\frac{1}{9}\))+(1-\(\frac{1}{16}\))+...+(1-\(\frac{1}{10000}\))
= 99 - (\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}\)+....+\(\frac{1}{100^2}\)) => 99 - A
Dễ thấy A>0 =>S < 99 (1)
Lại có A= \(\frac{1}{2^2}\)+\(\frac{1}{3^2}\)+\(\frac{1}{4^2}\)+...+\(\frac{1}{100^2}\)
=> A<\(\frac{1}{1.2}\)+\(\frac{1}{2.3}\)+\(\frac{1}{3.4}\)+...+\(\frac{1}{99.100}\)
=>A<1-\(\frac{1}{2}\)+\(\frac{1}{2}\)-\(\frac{1}{3}\)+\(\frac{1}{3}\)-\(\frac{1}{4}\)+...\(\frac{1}{99}\)-\(\frac{1}{100}\)
=>A<1-\(\frac{1}{100}\)<1
...
\(A=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}....\frac{9999}{10000}\)
\(=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}.....\frac{99.101}{100.100}\)
\(=\frac{\left(1.2.3....99\right)\left(3.4.5....101\right)}{\left(2.3.4...100\right)\left(2.3.4...100\right)}\)
\(=\frac{1.101}{100.2}=\frac{101}{200}\)
\(=\dfrac{1.3}{2^2}.\dfrac{2.4}{3^2}.\dfrac{3.5}{4^2}...\dfrac{99.101}{100^2}\)
\(=\dfrac{1.2...99}{2.3...100}.\dfrac{3.4...101}{2.3...100}=\dfrac{1}{100}.\dfrac{101}{2}=\dfrac{101}{200}\)
= 3 . 8 . 15 .... 9999 / 4 . 9 . 16 .... 10000
= ( 1 . 3 ) . ( 2 . 4 ) .( 3 . 5) .... ( 99 .... 101 ) / ( 2. 2) . (3.3). (4.4)...(100.100)
= 1. 101/100.2
= 101/ 200
k nha , đúng đó
1*3/2*2.2*4/3*3.3*5/4*4.....99*101/100*100. =1*2*3*...*99/2*3*4*...*100.3*4*5*...*101/2*3*4*...*100. =1/100 . 101/2. =101/200.
