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B= 3/2.4/3. ....2001/2000
B = 3.4....2001/2.3....2000
B =2001/2
Tính:\(A=\left(1-\dfrac{1}{4}\right)\left(1-\dfrac{1}{9}\right)\left(1-\dfrac{1}{16}\right)...\left(1-\dfrac{1}{10000}\right)\)\(=\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)\left(1-\dfrac{1}{4^2}\right)...\left(1-\dfrac{1}{100^2}\right)\)\(=\dfrac{3}{2^2}.\dfrac{2.4}{3^2}.\dfrac{3.5}{4^2}...\dfrac{99.101}{100^2}\)
\(=\dfrac{2.3.4...99}{2.3.4...100}.\dfrac{3.4.5.6...101}{2.3.4...100}\)
\(=\)\(\dfrac{1}{100}.\dfrac{101}{2}=\dfrac{101}{200}\)
Ta có thể làm cách khác:
(1+1/2)*(1+1/3)*(1+1/4)*...*(1+1/98)*(1+1/99)
= 3/2 x 4/3 x 5/4 x ... x 99/98 x 100/99 (giản ước ta được)
= 100/2 = 50
\(\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right)...\left(1+\frac{1}{99}\right)=\frac{3}{2}.\frac{4}{3}.\frac{5}{4}....\frac{100}{99}=\frac{100}{2}=50\)
A=1/21+1/22+1/23+...+1/40(có 20 phân số)
A<1/20+1/20+1/20+..+1/20(có 20 phân số)
A<20/20=1(1)
A>1/40+1/40+1/40+...+1/40(có 20 phân số)
A>20/40=1/2(2)
từ (1);(2) ta kết luận 1/2<A<1(câu 1)
dễ thấy A=.1/2+1/2^2+1/2^3+...+1/2^200
A<1/1*2+1/2*3+...+1/200*201
A<1/1-1/2+1/2-1/3+1/3-1/4+...+1/200-1/201
A<1-1/201<1
A<1
KL:0<A<1
A=\(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2013}-\frac{1}{2014}\)
A=\(\frac{1}{5.6}\)+\(\frac{1}{6.7}\)+\(\frac{1}{7.8}\)+\(\frac{1}{8.9}\)+\(\frac{1}{9.10}\)+\(\frac{1}{10.11}\)+\(\frac{1}{11.12}\)
=1/5-1/6+1/6-1/7+1/7-1/8+1/8-1/9+1/9-1/10+1/10-1/11+1/11-1/12
=1/5-1/12
=7/60
Dấu chấm là dấu nhân nhé bạn
A=1/30+1/42+1/56+1/72+1/90+1/110+1/132
A=1/5*6+1/6*7+1/7*8+1/8*9+1/9*10+1/10*11+1/11*12
A=1/5-1/6+1/6-1/7+1/7-1/8+1/8-1/9+1/9-1/10+1/10-1/11+1/11-1/12
A=1/5-1/12
A=7/60
Ta có, với \(n\) nguyên dương: \(1+2+...+n=\frac{n\left(n+1\right)}{2}\)
Suy ra, \(1-\frac{1}{1+2+...+n}=1-\frac{2}{n\left(n+1\right)}=\frac{\left(n-1\right)\left(n+2\right)}{n\left(n+1\right)}\)
Khi đó:
\(1-\frac{1}{1+2}=\frac{1.4}{2.3}\)
\(1-\frac{1}{1+2+3}=\frac{2.5}{3.4}\)
....
\(1-\frac{1}{1+2+...+2013}=\frac{2012.2015}{2013.2014}\)
\(1-\frac{1}{1+2+...+2014}=\frac{2013.2016}{2014.2015}\)
Suy ra, \(P=\frac{\left(1.2.....2013\right).\left(4.5.....2016\right)}{2.\left(3.4.....2014\right)^2.2015}=\frac{2016}{3.2014}=\frac{336}{1007}\)
1+1=2
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