CMR 1+1/2+1/3+.......+1/2^1999 >1000
Những câu hỏi liên quan
cmr :1+1/2+1/3+...+1/2 mũ 1999>1000
Câu hỏi: CMR: 1+1/2+1/3+....1/2^1999 >1000
CMR:1+1/2+1/3+...+1/21999>1000
C ={(1+(1999/1))(1+(1999/2))(1+(1999/3))+...+(1+(1999/1000))}/{(1+(1000/1))(1+(1000/2))(1+(1000/3))...(1+(1000/1999))}
A=(1+1999/1).(1+1992/2).(1+1999/3)...(1+1999/1000)/(1+1000/1).(1+1000/2).(1+1000/3)...(1+1000/1999)
Tính A
1.Tính C=\(\frac{\left(1+\frac{1999}{1}\right)\left(1+\frac{1999}{2}\right)\left(1+\frac{1999}{3}\right)...\left(1+\frac{1999}{1000}\right)}{\left(1+\frac{1000}{1}\right)\left(1+\frac{1000}{2}\right)\left(1+\frac{1000}{3}\right)...\left(1+\frac{1000}{1999}\right)}\)
\(C=\frac{\left(1+\frac{1999}{1}\right)\left(1+\frac{1999}{2}\right)...\left(1+\frac{1999}{1000}\right)}{\left(1+\frac{1000}{1}\right)\left(1+\frac{1000}{2}\right)...\left(1+\frac{1000}{1999}\right)}\)=> \(C=\frac{\frac{2000.2001.2002....2999}{1.2.3...1000}}{\frac{1001.1002.1003....2999}{1.2.3...1999}}\)
=> \(C=\frac{\frac{2000.2001.2002....2999}{1.2.3...1000}}{\frac{\left(1001.1002.1003....1999\right).\left(2000.2001.2002...2999\right)}{\left(1.2.3...1000\right).\left(1001.1002...1999\right)}}\)
=> \(C=\frac{2000.2001.2002....2999}{1.2.3...1000}.\frac{\left(1.2.3...1000\right).\left(1001.1002...1999\right)}{\left(1001.1002.1003....1999\right).\left(2000.2001.2002...2999\right)}=1\)
Đáp số: C=1
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Xem thêm câu trả lời
1)CMR :1+12+1/3+..+1/2^1999<1000
2)tím số tự nhiên x biết :1/3+1/3+1/10+...+1/x(x+1)=2013/2015
\(A=\frac{\left(1+\frac{1999}{1}\right)\left(1+\frac{1999}{2}\right)\left(1+\frac{1999}{3}\right)...\left(1+\frac{1999}{1000}\right)}{\left(1+\frac{1000}{1}\right)\left(1+\frac{1000}{2}\right)\left(1+\frac{1000}{3}\right)...\left(1+\frac{1000}{1999}\right)}\)
hỏi a = ?
A = (1 + 1999/1)(1 + 1999/2)......(1 + 1999/1000)
B = ( 1 + 1000/1)(1 + 1000/2)......(1 + 1000/1999)
Tính A/B
\(A=\left(1+\dfrac{1999}{1}\right)\left(1+\dfrac{1999}{2}\right)...\left(1+\dfrac{1999}{1000}\right)\)
\(=\dfrac{2000}{1}.\dfrac{2001}{2}.\dfrac{2002}{3}...\dfrac{2999}{1000}\)\(=\dfrac{2000.2001.2002...2999}{1.2.3...1000}\)
\(B=\left(1+\dfrac{1000}{1}\right)\left(1+\dfrac{1000}{2}\right)...\left(1+\dfrac{1000}{1999}\right)\)
\(=\dfrac{1001}{1}.\dfrac{1002}{2}.\dfrac{1003}{3}...\dfrac{2999}{1999}\) \(=\dfrac{1001.1002.1003...2999}{1.2.3...1999}\)
\(\Rightarrow A:B=\left(\dfrac{2000.2001.2002...2999}{1.2.3...1000}\right):\left(\dfrac{1001.1002.1003...2999}{1.2.3...1999}\right)\)
\(=\dfrac{2000.2001.2002...2999}{1.2.3...1000}.\dfrac{1.2.3...1999}{1001.1002.1003...2999}\)
\(=\dfrac{2000.2001.2002...2999}{1.2.3...1000}.\dfrac{1.2.3...1000.\left(1001.1002...1999\right)}{1001.1002.1003....1999.\left(2000.2001.2002.2999\right)}\)\(=\dfrac{1.2.3...1000}{1.2.3...1000}=1\)
Vậy \(\dfrac{A}{B}=1\)
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