Toán Tính S= (1- 1/2^2 )(1- 1/3^2 )…(1- 1/100^2 ) 24/11/2021 By Autumn Tính S= (1- 1/2^2 )(1- 1/3^2 )…(1- 1/100^2 )
$A = (1-\dfrac{1}{2^2}).(1-\dfrac{1}{3^2})….(1-\dfrac{1}{2020^2})$ Các số trên có dạng $1-\dfrac{1}{a^2} = \dfrac{a^2-1}{a^2} = \dfrac{(a-1).(a+1)}{a^2}$ Do đó $A = (1-\dfrac{1}{2^2}).(1-\dfrac{1}{3^2})….(1-\dfrac{1}{100^2})$ $ = \dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}.\dfrac{3.5}{4.4}….\dfrac{99.101}{100.100}$ $ = \dfrac{(1.2.3…99).(3.4…101)}{2^2.3^2.4^2….100^2}$ $ = \dfrac{101}{200}$ Trả lời
Đáp án: $S=\frac{101}{200}$ Giải thích các bước giải: $S= (1- \frac{1}{2^2} )(1- \frac{1}{3^2} )…(1- \frac{1}{100^2} )\\=(1-\frac{1}{4})(1-\frac{1}{9})….(1-\frac{1}{10000})\\=\frac{3}{4}.\frac{8}{9}.\frac{9999}{10000}\\=\frac{1.3}{2^2}.\frac{2.4}{3^2}….\frac{99.101}{100^2}\\=\frac{1.2.3…99}{2.3.4…100}.\frac{3.4.5…101}{2.3.4…100}\\=\frac{1}{100}.\frac{101}{2}\\=\frac{101}{200}$ Trả lời
$A = (1-\dfrac{1}{2^2}).(1-\dfrac{1}{3^2})….(1-\dfrac{1}{2020^2})$
Các số trên có dạng
$1-\dfrac{1}{a^2} = \dfrac{a^2-1}{a^2} = \dfrac{(a-1).(a+1)}{a^2}$
Do đó $A = (1-\dfrac{1}{2^2}).(1-\dfrac{1}{3^2})….(1-\dfrac{1}{100^2})$
$ = \dfrac{1.3}{2.2}.\dfrac{2.4}{3.3}.\dfrac{3.5}{4.4}….\dfrac{99.101}{100.100}$
$ = \dfrac{(1.2.3…99).(3.4…101)}{2^2.3^2.4^2….100^2}$
$ = \dfrac{101}{200}$
Đáp án:
$S=\frac{101}{200}$
Giải thích các bước giải:
$S= (1- \frac{1}{2^2} )(1- \frac{1}{3^2} )…(1- \frac{1}{100^2} )\\
=(1-\frac{1}{4})(1-\frac{1}{9})….(1-\frac{1}{10000})\\
=\frac{3}{4}.\frac{8}{9}.\frac{9999}{10000}\\
=\frac{1.3}{2^2}.\frac{2.4}{3^2}….\frac{99.101}{100^2}\\
=\frac{1.2.3…99}{2.3.4…100}.\frac{3.4.5…101}{2.3.4…100}\\
=\frac{1}{100}.\frac{101}{2}\\
=\frac{101}{200}$