Xét \(P=\sqrt{\frac{1}{1^2}+\frac{1}{a^2}+\frac{1}{\left(a+1\right)^2}}\) với a>0 

  \(P^2=1+\frac{1}{a^2}+\frac{1}{\left(a+1\right)^2}\) 

           \(=\frac{a^2\left(a+1\right)^2+\left(a+1\right)^2+a^2}{a^2\left(a+1\right)^2}\) 

           \(=\frac{a^2\left(a^2+2a+1+1\right)+\left(a+1\right)^2}{a^2\left(a+1\right)^2}\) 

           \(=\frac{a^4+2a^2\left(a+1\right)+\left(a+1\right)^2}{a^2\left(a+1\right)^2}\) 

           \(=\frac{\left(a^2+a+1\right)^2}{a^2\left(a+1\right)^2}\) 

           \(=\left(\frac{a^2+a+1}{a\left(a+1\right)}\right)^2\) 

Do a>o nên \(P=\frac{a^2+a+1}{a\left(a+1\right)}=1+\frac{1}{a}-\frac{1}{a+1}\) 

Áp dụng kết quả của P ta có:

 \(A=\left(1+\frac{1}{1}-\frac{1}{2}\right)+\left(1+\frac{1}{2}+\frac{1}{3}\right)+....+\left(1+\frac{1}{2012}-\frac{1}{2013}\right)\)      \(A=2012+\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-.....+\frac{1}{2012}-\frac{1}{2013}\right)\)  

\(A=2012+1-\frac{1}{2013}\)

\(A=2013-\frac{1}{2013}=\frac{4052168}{2013}\) 

Vậy \(A=\frac{4052168}{2013}\)

Mình giúp phần a thôi, phần b chir là áp dụng không có gì khó cả.

\(\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+2\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}\right)\)

\(=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+2\left(\frac{a+b+c}{abc}\right)=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\left(a+b+c=0\right)\)

\(\Rightarrow\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}=\sqrt{\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2}=\left|\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right|\left(đpcm\right)\)

b, \(A=\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+...+\sqrt{1+\frac{1}{399^2}+\frac{1}{400^2}}\)

\(A=\sqrt{\frac{1}{1^2}+\frac{1}{1^2}+\frac{1}{\left(-2\right)^2}}+\sqrt{\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{\left(-3\right)^2}}+...+\sqrt{\frac{1}{1^2}+\frac{1}{399^2}+\frac{1}{\left(-400\right)^2}}\)

có 1 + 1 - 2 = 1 + 2 - 3 = ... + 1 + 399 - 400 = 0

nên theo câu a ta có : 

\(A=\left|1+\frac{1}{1}-\frac{1}{2}\right|+\left|1+\frac{1}{2}-\frac{1}{3}\right|+...+\left|1+\frac{1}{399}-\frac{1}{400}\right|\)

A = 1 + 1 -1/2 + 1 + 1/2 - 1/3 + 1 + 1/3 - 1/4 + ... + 1 + 1/399 - 1/400

= 400  1/400

= 159999/400

Bạn ơi cho mình hỏi áp dụng như lào vậy???

\(A=3.\frac{1}{2}\left(2.\frac{1}{3}+\frac{-1}{3}\right)\)

\(A=\frac{3}{2}.\frac{1}{3}=\frac{1}{2}\)

\(B=\frac{-1}{2}\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}\right)\)

\(B=\frac{-1}{2}.\frac{1}{2}=-\frac{1}{4}\)

Tu lam di 

Sao hoi quai day

Phai biet suy nghi cho

bạn tìm trong nâng cao phát triển toán 9 tập 1 ấy nó có ở đấy

Bài 1:

A = \(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{49.50}\)

= \(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\)

= \(1-\frac{1}{50}=\frac{49}{50}\)

Bài 2:

Ta có: \(\frac{1}{1^2}=1;\frac{1}{2^2}< \frac{1}{1.2};\frac{1}{3^2}< \frac{1}{2.3};...;\frac{1}{50^2}< \frac{1}{49.50}\)

\(\Rightarrow A< 1+\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{49.50}\)

\(A< 1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}=1+1-\frac{1}{50}=2-\frac{1}{50}< 2\)

Vậy A < 2

Bài 3:

\(A=\frac{1}{30}+\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+\frac{1}{90}+\frac{1}{110}+\frac{1}{132}\)

\(=\frac{1}{5.6}+\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+\frac{1}{9.10}+\frac{1}{10.11}+\frac{1}{11.12}\)

\(=\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}+\frac{1}{9}-\frac{1}{10}+\frac{1}{10}-\frac{1}{11}+\frac{1}{11}-\frac{1}{12}\)

\(=\frac{1}{5}-\frac{1}{12}=\frac{7}{60}\)

Bài 4:

\(S=3+\frac{3}{2}+\frac{3}{2^2}+...+\frac{3}{2^9}\)

\(2S=6+3+\frac{3}{2}+...+\frac{3}{2^8}\)

\(2S-S=\left(6+3+\frac{3}{2}+...+\frac{3}{2^8}\right)-\left(3+\frac{3}{2}+\frac{3}{2^2}+...+\frac{3}{2^9}\right)\)

\(S=6-\frac{3}{2^9}=6-\frac{3}{512}=\frac{3069}{512}\)

A=1-1/2+1/2-1/3+.............................1/49-1/50

A=1-1/50

A=49/50

Bài 1 và bài 2 bạn tự làm nhé! Hoặc là search lên mạng tìm vì mik nhớ đã làm nhiều lần lắm rồi! Nhưng nếu bạn cần giải lại thì nhớ nói với mik nha

3)\(A=\frac{1}{30}+\frac{1}{42}+...+\frac{1}{132}\)
\(A=\frac{1}{5.6}+\frac{1}{6.7}+...+\frac{1}{11.12}\)
\(A=\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{11}-\frac{1}{12}\)
\(A=\frac{1}{5}-\frac{1}{12}=\frac{7}{60}\)
4)\(S=3+\frac{3}{2}+\frac{3}{2^2}+...+\frac{3}{2^9}\)
\(\Rightarrow\frac{1}{2}S=\frac{3}{2}+\frac{3}{2^2}+\frac{3}{2^3}+...+\frac{3}{2^{10}}\)
\(\Rightarrow S=\left(s-\frac{1}{2}S\right).2=\left(3-\frac{3}{2^{10}}\right).2=\frac{3069}{512}\)

A = \(\frac{1}{2}\)\(-\)\(\frac{1}{2^2}\)\(+\)\(\frac{1}{2^3}\)\(-\)\(\frac{1}{2^4}\)\(+\)........... \(+\)\(\frac{1}{2^{99}}\)\(-\)\(\frac{1}{2^{100}}\)

2A = 1 - \(\frac{1}{2}\)+ \(\frac{1}{2^2}\)- \(\frac{1}{2^3}\)+.........+ \(\frac{1}{2^{98}}\)- \(\frac{1}{2^{99}}\)

2A + A =( 1 - \(\frac{1}{2}\)+ \(\frac{1}{2^2}\)- \(\frac{1}{2^3}\)+.........+ \(\frac{1}{2^{98}}\)- \(\frac{1}{2^{99}}\)) \(+\)( \(\frac{1}{2}\)\(-\)\(\frac{1}{2^2}\)\(+\)\(\frac{1}{2^3}\)\(-\)\(\frac{1}{2^4}\)\(+\)........... \(+\)\(\frac{1}{2^{99}}\)\(-\)\(\frac{1}{2^{100}}\)) 

3A = 1 \(-\) \(\frac{1}{2^{100}}\)

\(\Rightarrow\)A = \(\frac{1-\frac{1}{2^{100}}}{3}\)= \(\frac{1}{3}\)

Bài này có 2 cách!!

\(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}=\)\(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{2\left(a+b+c\right)}{abc}}\)=\(\sqrt{\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^2}=\)\(|\frac{1}{a}+\frac{1}{b}+\frac{1}{c}|\)

\(\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}=\sqrt{1+\frac{1}{2^2}+\frac{1}{\left(-3\right)^2}}\)\(=|\frac{1}{1}+\frac{1}{2}+\frac{1}{-3}|=1+\frac{1}{2}-\frac{1}{3}\)

Tương tự ta có M=\(1+\frac{1}{2}-\frac{1}{3}+1+\frac{1}{3}-\frac{1}{4}+...+1+\frac{1}{99}-\frac{1}{100}\)=\(98+\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{99}\right)-\left(\frac{1}{3}+\frac{1}{4}+...+\frac{1}{100}\right)\)\(=98+\frac{1}{2}-\frac{1}{100}=\frac{9849}{100}\)