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a) Trục căn thức ở mỗi số hạng của biểu thức A,ta có:
\(A=\frac{1}{\sqrt{1}-\sqrt{2}}-\frac{1}{\sqrt{2}-\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{4}}-...+\frac{1}{\sqrt{2007}-\sqrt{2008}}\)=\(\frac{\sqrt{2}+\sqrt{1}}{1-2}-\frac{\sqrt{3}+\sqrt{2}}{2-3}+\frac{\sqrt{3}+\sqrt{4}}{3-4}-...+\frac{\sqrt{2007}+\sqrt{2008}}{2007-2008}\)
= \(-\left(\sqrt{1}+\sqrt{2}\right)+\left(\sqrt{2}+\sqrt{3}\right)-\left(\sqrt{3}+\sqrt{4}\right)+...-\left(\sqrt{2007}+\sqrt{2008}\right)\)
=\(-1-\sqrt{2008}\)
b)Ta xét số hạng tổng quát: \(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}\)=\(\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{\left(n+1\right)^2n-n^2\left(n+1\right)}\)=\(\frac{\left(n+1\right)\sqrt{n}-n\sqrt{n+1}}{n\left(n+1\right)}\)=\(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)
Áp dụng vào biểu thức B ta được:
B= \(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}-...+\frac{1}{\sqrt{120}}-\frac{1}{\sqrt{121}}=1-\frac{1}{11}\)= \(\frac{10}{11}\)
\(A=\frac{1}{\sqrt{1}-\sqrt{2}}-\frac{1}{\sqrt{2}-\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{4}}-\frac{1}{\sqrt{4}-\sqrt{5}}+...+\frac{1}{\sqrt{2007}-\sqrt{2008}}\)
\(=\frac{-1}{\sqrt{2}-\sqrt{1}}+\frac{1}{\sqrt{3}-\sqrt{2}}-\frac{1}{\sqrt{4}-\sqrt{3}}+\frac{1}{\sqrt{5}-\sqrt{4}}-....+\frac{1}{\sqrt{2007}-\sqrt{2006}}-\frac{1}{\sqrt{2008}-\sqrt{2007}}\)
\(=\frac{-1\cdot\left(\sqrt{2}+\sqrt{1}\right)}{2-1}+\frac{1\cdot\left(\sqrt{3}+\sqrt{2}\right)}{3-2}-\frac{1\cdot\left(\sqrt{4}+\sqrt{3}\right)}{4-3}+\frac{1\cdot\left(\sqrt{5}+\sqrt{4}\right)}{5-4}-...+\frac{1\cdot\left(\sqrt{2007}+\sqrt{2006}\right)}{2007-2006}-\frac{1 \left(\sqrt{2008}+\sqrt{2007}\right)}{2008-2007}\)
\(=-1-\sqrt{2}+\sqrt{2}+\sqrt{3}-\sqrt{3}-\sqrt{4}+\sqrt{4}+\sqrt{5}-...+\sqrt{2006}+\sqrt{2007}-\sqrt{2007}-\sqrt{2008}\)
\(=-1-\sqrt{2008}\)
Lời giải:
\(A=\sqrt{1+\frac{1}{a^2}+\frac{1}{(a+1)^2}}=\sqrt{1+2.\frac{1}{a}+\frac{1}{a^2}+\frac{1}{(a+1)^2}-\frac{2}{a}}\)
\(=\sqrt{(1+\frac{1}{a})^2+\frac{1}{(a+1)^2}-\frac{2}{a}}=\sqrt{\frac{(a+1)^2}{a^2}+\frac{1}{(a+1)^2}-2.\frac{a+1}{a}.\frac{1}{a+1}}\)
\(=\sqrt{(\frac{a+1}{a}-\frac{1}{a+1})^2}=|\frac{a+1}{a}-\frac{1}{a+1}|=|1+\frac{1}{a}-\frac{1}{a+1}|\)
b)
Áp dụng công thức trên vào bài toán:
\(B=\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+....+\sqrt{1+\frac{1}{99^2}+\frac{1}{100^2}}\)
\(=|1+\frac{1}{1}-\frac{1}{2}|+|1+\frac{1}{2}-\frac{1}{3}|+....+|1+\frac{1}{99}-\frac{1}{100}|\)
\(=99+(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+....+\frac{1}{99}-\frac{1}{100})\)
\(=99+1-\frac{1}{100}=100-\frac{1}{100}\)
Sai đề nha bn \(A=\sqrt{1+\frac{1}{a^2}+\frac{1}{\left(a+1\right)^2}}\)
\(A=\sqrt{\frac{a^2\left(a+1\right)^2+\left(a+1\right)^2+a^2}{a^2\left(a+1\right)^2}}\)\(=\sqrt{\frac{a^2\left(a+1\right)^2+2a^2+2a+1}{a^2\left(a+1\right)^2}}\)
\(=\sqrt{\frac{\left[a\left(a+1\right)^2\right]+2a\left(a+1\right)+1}{a^2\left(a+1\right)^2}}\) \(=\sqrt{\frac{\left[a\left(a+1\right)+1\right]^2}{a^2\left(a+1\right)^2}}\)
\(=\frac{a\left(a+1\right)+1}{a\left(a+1\right)}=1+\frac{1}{a\left(a+1\right)}=1+\frac{1}{a}-\frac{1}{a+1}\)
Áp dụng kết quả trên ta có :
\(B=1+1-\frac{1}{2}+1+\frac{1}{2}-\frac{1}{3}+...+1+\frac{1}{99}-\frac{1}{100}\)
\(=99+1-\frac{1}{100}=\frac{9999}{100}\)
Có: \(\left(\frac{1}{\sqrt{\frac{9}{4}+\sqrt{5}}}-\frac{1}{\sqrt{\frac{9}{4}-\sqrt{5}}}\right)^2\)
\(=\frac{1}{\frac{9}{4}+\sqrt{5}}+\frac{1}{\frac{9}{4}-\sqrt{5}}-2\cdot\frac{1}{\sqrt{\frac{9}{4}+\sqrt{5}}}\cdot\frac{1}{\sqrt{\frac{9}{4}-\sqrt{5}}}\)
\(=\frac{\frac{9}{4}-\sqrt{5}+\frac{9}{4}+\sqrt{5}}{\frac{1}{16}}-2\cdot\frac{1}{\frac{1}{4}}\)
\(=72-8=64\)
Mà; \(\frac{1}{\sqrt{\frac{9}{4}+\sqrt{5}}}< \frac{1}{\sqrt{\frac{9}{4}-\sqrt{5}}}\)
\(\Rightarrow\frac{1}{\sqrt{\frac{9}{4}+\sqrt{5}}}-\frac{1}{\sqrt{\frac{9}{4}-\sqrt{5}}}< 0\)
Do đó: \(\frac{1}{\sqrt{\frac{9}{4}+\sqrt{5}}}-\frac{1}{\sqrt{\frac{9}{4}-\sqrt{5}}}=-8\)
Khi đó: \(x=9-8=1\)
Với \(x=1\), ta có:
\(f\left(1\right)=\left(1^4-3\cdot1+1\right)^{2016}=\left(-1\right)^{2016}=1\)
a, \(S=\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{2015.2017}\)
\(\Rightarrow\) \(2S=\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+...+\frac{2}{2015.2017}\)
\(\Rightarrow\) \(2S=1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{2015}-\frac{1}{2017}\)
\(\Rightarrow\) \(2S=1-\frac{1}{2017}\)
\(\Rightarrow\) \(2S=\frac{2016}{2017}\)
\(\Rightarrow\) \(S=\frac{1008}{2017}\)
Bạn hãy chứng minh đẳng thức phụ sau : \(\sqrt{1+\frac{1}{\left(k-1\right)^2}+\frac{1}{k^2}}=\left|1-\frac{1}{k}+\frac{1}{k+1}\right|\)
Áp dụng : \(\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+...+\sqrt{1+\frac{1}{99^2}+\frac{1}{100^2}}=\left(1+1-\frac{1}{2}\right)+\left(1+\frac{1}{2}-\frac{1}{3}\right)+...+\left(1+\frac{1}{99}-\frac{1}{100}\right)\)\(=1.99+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{100}=100-\frac{1}{100}\)
Với a \(\in\)N*, ta có:
\(\sqrt{1+\frac{1}{a^2}+\frac{1}{\left(a+1\right)^2}}=\sqrt{\frac{a^2.\left(a+1\right)^2}{a^2.\left(a+1\right)^2}+\frac{\left(a+1\right)^2}{a^2.\left(a+1\right)^2}+\frac{a^2}{a^2.\left(a+1\right)^2}}\)
\(=\sqrt{\frac{\left[a.\left(a+1\right)\right]^2+\left(a+1\right)^2+a^2}{\left[a.\left(a+1\right)\right]^2}}=\sqrt{\frac{\left[a.\left(a+1\right)\right]^2+a^2+2a+1+a^2}{\left[a.\left(a+1\right)\right]^2}}\)
\(=\sqrt{\frac{\left[a.\left(a+1\right)\right]^2+2a^2+2a+1}{\left[a.\left(a+1\right)\right]^2}}=\sqrt{\frac{\left[a.\left(a+1\right)\right]^2+2.\left(a^2+a\right)+1}{\left[a.\left(a+1\right)\right]^2}}\)
\(=\sqrt{\frac{\left[a.\left(a+1\right)\right]^2+2.a.\left(a+1\right).1+1^2}{\left[a.\left(a+1\right)\right]^2}}=\sqrt{\frac{\left[a.\left(a+1\right)+1\right]^2}{\left[a.\left(a+1\right)\right]^2}}\)
\(=\sqrt{\left[\frac{a.\left(a+1\right)+1}{a.\left(a+1\right)}\right]^2}=\frac{a.\left(a+1\right)+1}{a.\left(a+1\right)}=\frac{a.\left(a+1\right)}{a.\left(a+1\right)}+\frac{1}{a.\left(a+1\right)}\)
\(=1+\frac{a+1-a}{a.\left(a+1\right)}=1+\frac{a+1}{a.\left(a+1\right)}-\frac{a}{a.\left(a+1\right)}=a+\frac{1}{a}-\frac{1}{a+1}\)
=>\(\sqrt{1+\frac{1}{a^2}+\frac{1}{\left(a+1\right)^2}}=1+\frac{1}{a}-\frac{1}{a+1}\)
Thay a=1,2,...99
=>\(\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}=1+1-\frac{1}{2}\)
\(\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}=1+\frac{1}{2}-\frac{1}{3}\)
............................................................
\(\sqrt{1+\frac{1}{99^2}+\frac{1}{100^2}}=1+\frac{1}{99}-\frac{1}{100}\)
=>\(\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+...+\sqrt{1+\frac{1}{99^2}+\frac{1}{100^2}}\)
\(=1+1-\frac{1}{2}+1+\frac{1}{2}-\frac{1}{3}+...+1+\frac{1}{99}-\frac{1}{100}\)
\(=1+1+...+1-\frac{1}{100}\)
\(=100-\frac{1}{100}\)
\(=\frac{9999}{100}\)
Vậy \(\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+...+\sqrt{1+\frac{1}{99^2}+\frac{1}{100^2}}=\frac{9999}{100}\)
Với \(k\in N;k\ne0\) ta có :
\(\frac{1}{\left(k+1\right)\sqrt{k}+k\sqrt{\left(k+1\right)}}=\frac{1}{\sqrt{k\left(k+1\right)}\left(\sqrt{k}+\sqrt{k+1}\right)}\)
\(=\frac{\sqrt{k+1}+\sqrt{k}}{\sqrt{k\left(k+1\right)}\left(\sqrt{k+1}-\sqrt{k}\right)\left(\sqrt{k+1}+\sqrt{k}\right)}=\frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k\left(k+1\right)}}\)
\(=\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}\)
Áp dụng ta có :
\(M=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+....+\frac{1}{\sqrt{120}}-\frac{1}{\sqrt{121}}=1-\frac{1}{11}=\frac{10}{11}\)
ta có
\(S=\frac{1}{10}+\frac{1}{20}+\frac{1}{35}+\frac{1}{56}+\frac{1}{84}+\frac{1}{120}+\frac{1}{165}+\frac{1}{220}\)
\(=6\left(\frac{1}{3\cdot4\cdot5}+\frac{1}{4\cdot5\cdot6}+\frac{1}{6\cdot7\cdot8}+\frac{1}{8\cdot9\cdot10}+\frac{1}{10\cdot11\cdot12}\right)\)
\(=3\left(\frac{1}{3\cdot4}-\frac{1}{11\cdot12}\right)=\frac{5}{22}\)