Question

a. Rút gọn biểu thức \(A=\sqrt{1+\frac{1}{a^2}+\frac{1}{\left(1+a^2\right)}}\)

b. Tính giá trị của tổng \(B=\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+\sqrt{1+\frac{1}{3^2}+\frac{1}{4^2}}+...+\sqrt{1+\frac{1}{99^2}+\frac{1}{100^2}}\)

Cảm ơn mọi người nhiều

Answer 1

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:

\(B=\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+....+\sqrt{1+\frac{1}{99^2}+\frac{1}{100^2}}\)

\(=|1+\frac{1}{1}-\frac{1}{2}|+|1+\frac{1}{2}-\frac{1}{3}|+....+|1+\frac{1}{99}-\frac{1}{100}|\)

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

\(=99+1-\frac{1}{100}=100-\frac{1}{100}\)

Answer 2

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 :

\(B=1+1-\frac{1}{2}+1+\frac{1}{2}-\frac{1}{3}+...+1+\frac{1}{99}-\frac{1}{100}\)

\(=99+1-\frac{1}{100}=\frac{9999}{100}\)