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Xét : Với mọi \(x\in N^{\text{*}}\) , ta có : \(\frac{1}{\left(x+1\right)\sqrt{x}+x\sqrt{x+1}}=\frac{1}{\sqrt{x\left(x+1\right)}\left(\sqrt{x}+\sqrt{x+1}\right)}=\frac{\sqrt{x+1}-\sqrt{x}}{\sqrt{x\left(x+1\right)}}=\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x+1}}\)
Áp dụng vào tính : \(M=\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+...+\frac{1}{100\sqrt{99}+99\sqrt{100}}=1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{99}}-\frac{1}{\sqrt{100}}=1-\frac{1}{10}=\frac{9}{10}\)
Với mọi n thuộc N ta có :
\(\sqrt{\frac{1}{1^2}+\frac{1}{n^2}+\frac{1}{\left(n+1\right)^2}}=\sqrt{1+\frac{1}{n^2}+\frac{1}{\left(n+1\right)^2}+\frac{2}{n}-\frac{2}{n\left(n+1\right)}-\frac{2}{\left(n+1\right)}}\)
\(=\sqrt{\left(1+\frac{1}{n}-\frac{1}{n+1}\right)^2}=1+\frac{1}{n}-\frac{1}{n+1}\)
Áp dụng ta được :
\(S=\left(1+\frac{1}{2}-\frac{1}{3}\right)+\left(1+\frac{1}{3}-\frac{1}{4}\right)+....+\left(1+\frac{1}{99}-\frac{1}{100}\right)\)
\(=98+\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\right)\)
\(=98+\frac{1}{2}-\frac{1}{100}=\frac{9849}{100}\)
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 :
\(B=1+1-\frac{1}{2}+1+\frac{1}{2}-\frac{1}{3}+...+1+\frac{1}{99}-\frac{1}{100}\)
\(=99+1-\frac{1}{100}=\frac{9999}{100}\)
Ta có \(A=\sqrt{1+\frac{1}{a^2}+\frac{1}{\left(a+1\right)^2}}\Rightarrow A^2=1+\frac{1}{a^2}+\frac{1}{\left(a+1\right)^2}=\frac{a^2\left(a+1\right)^2+\left(a+1\right)^2+a^2}{a^2\left(a+1\right)^2}\)
\(=\frac{a^2\left(a^2+2a+1+1\right)+\left(a+1\right)^2}{a^2\left(a+1\right)^2}=\frac{a^4+2a^2\left(a+1\right)+\left(a+1\right)^2}{a^2\left(a+1\right)^2}\)
= \(\frac{\left(a^2+a+1\right)^2}{a^2\left(a+1\right)^2}=\left[\frac{a^2+a+1}{a\left(a+1\right)}\right]^2\Rightarrow A=\frac{a^2+a+1}{a\left(a+1\right)}=1+\frac{1}{a\left(a+1\right)}\)
= \(1+\frac{1}{a}-\frac{1}{a+1}\)
rồi bạn thay vào tổng trên là xong
A = \(\frac{\sqrt{3+\sqrt{5}}}{\sqrt{2}}-\frac{\sqrt{5}-1}{2}=\frac{\sqrt{3+\sqrt{5}}.\sqrt{2}}{2}-\frac{\sqrt{5}-1}{2}\)
= \(\frac{\sqrt{5}+1}{2}-\frac{\sqrt{5}-1}{2}=1\)
\(B=\frac{\sqrt{2}-1}{2-1}+\frac{\sqrt{3}-\sqrt{2}}{3-2}+\frac{\sqrt{4}-\sqrt{3}}{4-3}+...+\frac{\sqrt{100}-\sqrt{99}}{100-99}\)
\(B=\sqrt{2}-1+\sqrt{3}-\sqrt{2}+\sqrt{4}-\sqrt{3}+...+\sqrt{100}-\sqrt{99}=-1+\sqrt{100}=10-1=9\)
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 \(\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+...+\sqrt{1+\frac{1}{99^2}+\frac{1}{100^2}}=\frac{9999}{100}\)
Ta xét biểu thức sau :
\(\frac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}=\frac{1}{\sqrt{n\left(n+1\right)}\left(\sqrt{n}+\sqrt{n+1}\right)}=\frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n\left(n+1\right)}\left[\left(\sqrt{n+1}\right)^2-\left(\sqrt{n}\right)^2\right]}\)
\(=\frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n\left(n+1\right)}}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)(với n > 0)
Áp dụng : \(\frac{1}{2\sqrt{1}+1\sqrt{2}}+\frac{1}{3\sqrt{2}+2\sqrt{3}}+...+\frac{1}{100\sqrt{99}+99\sqrt{100}}\)
\(=\left(\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}\right)+\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)+...+\left(\frac{1}{\sqrt{99}}-\frac{1}{\sqrt{100}}\right)\)
\(=1-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{99}}-\frac{1}{\sqrt{100}}=1-\frac{1}{\sqrt{100}}=1-\frac{1}{10}=\frac{9}{10}\)
ừm đúng rồi vậy sửa lại nha
A=100-1/100
A=99,99
Với mọi \(n\in N\)ta có:
\(\sqrt{1+\frac{1}{n^2}+\frac{1}{\left(n+1\right)^2}}\)
=\(\sqrt{1+\frac{1}{n^2}+\frac{1}{\left(n+1\right)^2}-\frac{2}{n}-\frac{2}{n\left(n+1\right)}-\frac{2}{n+1}}\)
\(=\sqrt{\left(1+\frac{1}{n}-\frac{1}{n+1}^2\right)}=1+\frac{1}{n}-\frac{1}{n+1}\)
Áp dung vào biểu thức A ta được:
\(A=\left(1+\frac{1}{1}-\frac{1}{2}\right)+\left(1+\frac{1}{2}-\frac{1}{3}\right)+....+\left(1+\frac{1}{99}-\frac{1}{100}\right)\)
\(A=1+1-\frac{1}{2}+1+\frac{1}{2}-\frac{1}{3}+...+1+\frac{1}{99}-\frac{1}{100}\)
\(A=\left(1+1+1+...+1\right)+\left(-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-....-\frac{1}{99}+\frac{1}{99}-\frac{1}{100}\right)\)(Có 99 số 1)
\(A=99-\frac{1}{100}=\frac{9899}{100}\)