Đề bài

Tính $A = \left( {1 - \frac{1}{{{2^2}}}} \right)\left( {1 - \frac{1}{{{3^2}}}} \right) \cdot  \cdot  \cdot \left( {1 - \frac{1}{{{{2010}^2}}}} \right)$.

• A. $\frac{{2009}}{{2010}}$
• B. $\frac{{2011}}{{2010}}$
• C. $\frac{{2011}}{{4020}}$
• D. $\frac{{2009}}{{4020}}$

Phương pháp giải

Sử dụng công thức $\left( {1 - \frac{1}{{{2^2}}}} \right)\left( {1 - \frac{1}{{{3^2}}}} \right)...\left( {1 - \frac{1}{{{n^2}}}} \right) = \frac{{n + 1}}{{2n}}$.

Lời giải của GV Loigiaihay.com

$\begin{array}{l}\left( {1 - \frac{1}{{{2^2}}}} \right)\left( {1 - \frac{1}{{{3^2}}}} \right)\left( {1 - \frac{1}{{{4^2}}}} \right)\left( {1 - \frac{1}{{{5^2}}}} \right) \cdot  \cdot  \cdot \left( {1 - \frac{1}{{{n^2}}}} \right) = \frac{{{2^2} - 1}}{{{2^2}}} \cdot \frac{{{3^2} - 1}}{{{3^2}}} \cdot \frac{{{4^2} - 1}}{{{4^2}}} \cdot \frac{{{5^2} - 1}}{{{5^2}}} \cdot  \cdot  \cdot \frac{{{n^2} - 1}}{{{n^2}}}\\ = \frac{{1.3}}{{{2^2}}} \cdot \frac{{2.4}}{{{3^2}}} \cdot \frac{{3.5}}{{{4^2}}} \cdot \frac{{4.6}}{{{5^2}}} \cdot  \cdot  \cdot \frac{{\left( {n - 1} \right)\left( {n + 1} \right)}}{{{n^2}}} = \frac{{1.2.3.4...\left( {n - 1} \right)}}{{2.3.4.5...n}} \cdot \frac{{3.4.5.6...\left( {n + 1} \right)}}{{2.3.4.5...n}} = \frac{1}{n} \cdot \frac{{n + 1}}{2} = \frac{{n + 1}}{{2n}}\end{array}$Áp dụng với $n = 2010$ ta có:

$A = \left( {1 - \frac{1}{{{2^2}}}} \right)\left( {1 - \frac{1}{{{3^2}}}} \right) \cdot  \cdot  \cdot \left( {1 - \frac{1}{{{{2010}^2}}}} \right) = \frac{{2010 + 1}}{{2.2010}} = \frac{{2011}}{{4020}}$

Đáp án : C