\(\frac{1+\frac{1}{3}+\frac{1}{5}+...+\frac{1}{2011}}{\frac{1}{2011\cdot1}+\frac{1}{3\cdot2009}+...+\frac{1}{2011\cdot1}}\)
Tính tổng S = \(\frac{1}{2011\cdot2009}-\frac{1}{2009\cdot2007}-...-\frac{1}{3\cdot1}\)
Ta có: 1/2011.2009 = 1/2011 - 1/2009 ; 1/2009.2007 = 1/2009 - 1/2007. làm tường tự vội số còn lại.
Ta có: 1/2011 - 1/2009 - 1/2009 - 1/2007- .....- 1/3 - 1/1
( đợi tối mình post tiếp)
Cho biểu thức \(C=\frac{1}{\sqrt{1\cdot2010}}+\frac{1}{\sqrt{2\cdot2009}}+\frac{1}{\sqrt{3\cdot2008}}+...+\frac{1}{\sqrt{2010\cdot1}}\)
So sánh C với \(D=2\cdot\frac{2010}{2011}\)
(1 +2010) > 2\(\sqrt{1.2010}\)=> \(\frac{1}{\sqrt{1.2010}}\)> 2/2011 tương tự các phần tử còn lại
vậy C > 2/2011+2/2011+.....2/2011 = 2.2010/2011
\(1\frac{1}{9}\cdot1\frac{1}{10}\cdot1\frac{1}{11}\cdot...\cdot1\frac{1}{2011}\)
dấu chấm ở giữa kia là dấu nhân
1 1/9 x 1 1/10 x 1 1/11 x ... x 1 1/2011
=10/9 x 11/10 x 12/11 x ... x 2012/2011
khử
còn 2012/9
=\(\frac{10}{9}\)x\(\frac{11}{10}\)x\(\frac{12}{11}\)x.........x\(\frac{2012}{2011}\)
=\(\frac{2012}{9}\)
\(1\frac{1}{9}.1\frac{1}{10}.1\frac{1}{11}...1\frac{1}{2011}\)
= \(\frac{10}{9}.\frac{11}{10}.\frac{12}{11}...\frac{2012}{2011}\)
= \(\frac{2012}{9}\)
\(\frac{45\cdot16-17}{45\cdot15+28}\) các bạn đề là tính nhanh nhé
Ví dụ: \(\frac{2016\cdot2015-5}{2014\cdot2016+2011}=\frac{2016\cdot\left(2014+1\right)-5}{2014\cdot2016+2011}=\frac{2016\cdot2014+2016\cdot1-5}{2014\cdot2016-2011}=\frac{2016\cdot2014+2011}{2014\cdot2016+2011}=1\)
rút gọn phân số \(1\frac{1}{3}\cdot1\frac{1}{8}\cdot1\frac{1}{15}\cdot1\frac{1}{24}\cdot...\cdot1\frac{1}{360}\)
1<1/3+1/8+1/15+1/24+....+1/360>
KO BIẾT ĐÚNG HAY KO NHÉ BẠN
\(1\frac{1}{3}.1\frac{1}{8}.1\frac{1}{15}.1\frac{1}{24}...1\frac{1}{360}\)
\(=\frac{4}{3}.\frac{9}{8}.\frac{16}{15}.\frac{25}{24}...\frac{361}{360}\)
\(=\frac{2^2}{3}.\frac{3^2}{8}.\frac{4^2}{15}.\frac{5^2}{24}...\frac{19^2}{360}\)
\(=\frac{2.2}{1.3}.\frac{3.3}{2.4}.\frac{4.4}{3.5}.\frac{5.5}{4.6}...\frac{19.19}{18.20}\)
\(=\left(\frac{2.3.4.5...19}{1.2.3.4...18}\right).\left(\frac{2.3.4.5...19}{3.4.5.6...20}\right)\)
\(=19.\frac{1}{10}\)
\(=\frac{19}{10}\)
Rút gọn:
\(A=1\frac{1}{3}\cdot1\frac{1}{8}\cdot1\frac{1}{15}\cdot1\frac{1}{24}\cdot...\cdot1\frac{1}{360}\)
\(A=1\frac{1}{3}.1\frac{1}{8}.1\frac{1}{15}.1\frac{1}{24}.....1\frac{1}{360}\)
\(A=1+\left(\frac{1}{3}.\frac{1}{8}.\frac{1}{15}.\frac{1}{24}.....\frac{1}{360}\right)\)
Nếu đúng thì tk nha
k=\(\frac{1+\left(1+2\right)+\left(1+2+3\right)+...+\left(1+2+3+...+2013\right)}{2013\cdot1+2012\cdot2+2011\cdot3+...+2\cdot2012+1\cdot2013}\)
1) Tính nhanh:
P=\(1\frac{1}{3}\cdot1\frac{1}{8}\cdot1\frac{1}{15}\cdot1\frac{1}{24}\cdot1\frac{1}{35}\cdot1\frac{1}{48}\cdot1\frac{1}{63}\cdot1\frac{1}{80}\)
2) So sánh:
A=\(\frac{100^{10}+1}{100^{10}-1}\) và B=\(\frac{100^{10}-1}{100^{10}-3}\)
3) So sánh A và B biết:
A=\(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{49\cdot50}\)
B=\(\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+...+\frac{1}{99}+\frac{1}{100}\)
#It's the moment when you're in good mood, you accidentally click back =.=
1) Calculate
\(P=1\frac{1}{3}.1\frac{1}{8}.1\frac{1}{15}....1\frac{1}{63}.1\frac{1}{80}\)
\(=\frac{4}{3}.\frac{9}{8}.\frac{16}{15}....\frac{64}{63}.\frac{81}{80}\)
\(=\frac{2.2}{1.3}.\frac{3.3}{2.4}.\frac{4.4}{3.5}....\frac{8.8}{7.9}.\frac{9.9}{8.10}\)
\(=\frac{2.9}{10}=\frac{9}{5}\)
ta có: 10010 + 1 > 10010 - 1
⇒ A = \(\frac{100^{10}+1}{100^{10}-1}< \frac{100^{10}+1-2}{100^{10}-1-2}=\frac{100^{10}-1}{100^{10}-3}=B\)
vậy A < B
3)
\(A=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)
\(=\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+...+\frac{50-49}{49.50}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)
\(=1-\frac{49}{50}\)
\(=\frac{49}{50}\)
⇒ A < 1 (1)
\(B=\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+...+\frac{1}{99}+\frac{1}{100}\)
\(\Rightarrow B>\frac{1}{10}+\left(\frac{1}{100}+\frac{1}{100}+...+\frac{1}{100}\right)=\frac{1}{10}+\frac{90}{100}=1\)
⇒ B > 1 (2)
từ (1) và (2) ⇒ A<1<B
vậy A < B
\(1\frac{1}{3}\cdot1\frac{1}{8}\cdot1\frac{1}{15}\cdot..........\cdot1\frac{1}{9800}\)tinh nhanh