\(A=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}....\frac{9999}{10000}\)

\(=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}.....\frac{99.101}{100.100}\)

\(=\frac{\left(1.2.3....99\right)\left(3.4.5....101\right)}{\left(2.3.4...100\right)\left(2.3.4...100\right)}\)

\(=\frac{1.101}{100.2}=\frac{101}{200}\)

3/4.8/9.15/16...9999/10000

=\(\dfrac{1.3}{2.2}\).\(\dfrac{2.4}{3.3}\)...\(\dfrac{99.101}{100.100}\)

=\(\dfrac{1.2...99}{2.3.100}\).\(\dfrac{3.4...101}{2.3.100}\)

=\(\dfrac{1}{100}\).\(\dfrac{101}{2}\)

=\(\dfrac{101}{200}\)

Lời giải:

$A=(1-\frac{1}{4})+(1-\frac{1}{9})+(1-\frac{1}{16})+....+(1-\frac{1}{10000})$

$=(1+1+...+1)-(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+....+\frac{1}{10000})$

$=99-(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+....+\frac{1}{10000})< 99$

Đề thiếu. Bạn xem lại đề.

\(\frac{3}{4}+\frac{8}{9}+\frac{15}{16}\)+...+\(\frac{9999}{10000}\)

= (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

...

Hzzz tự làm ko nổi

\(=\dfrac{1.3}{2^2}.\dfrac{2.4}{3^2}.\dfrac{3.5}{4^2}...\dfrac{99.101}{100^2}\)

\(=\dfrac{1.2...99}{2.3...100}.\dfrac{3.4...101}{2.3...100}=\dfrac{1}{100}.\dfrac{101}{2}=\dfrac{101}{200}\)

= 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.