Đáp án: $ A=\dfrac{201}{10100}$
Giải thích các bước giải:
Ta có:
$A=202(200^{-2}-1)(199^{-2}-1)(198^{-2}-1)...(101^{-2}-1)$
$\to A=202(\dfrac{1}{200^2}-1)(\dfrac{1}{199^2}-1)...(\dfrac{1}{101^2}-1)$
$\to A=202(1-\dfrac{1}{200^2})(1-\dfrac{1}{199^2})...(1-\dfrac{1}{101^2})$
$\to A=202\cdot \dfrac{200^2-1}{200^2}\cdot \dfrac{199^2-1}{199^2}....\dfrac{101^2-1}{101^2}$
$\to A=202\cdot \dfrac{(200-1)(200+1)}{200\cdot 200}\cdot \dfrac{(199-1)(199+1)}{199\cdot 199}....\dfrac{(101-1)(101+1)}{101\cdot 101}$
$\to A=202\cdot \dfrac{199\cdot 201}{200\cdot 200}\cdot \dfrac{198\cdot200}{199\cdot 199}....\dfrac{100\cdot102}{101\cdot 101}$
$\to A=\dfrac{199\cdot 198\cdot ...\cdot 100}{200\cdot 199...101}\cdot \dfrac{202\cdot 201\cdot ...102}{200\cdot 199\cdot ...\cdot 101}$
$\to A=\dfrac{1}{200\cdot 101}\cdot 2\cdot 201$
$\to A=\dfrac{1}{100\cdot 101}\cdot 201$
$\to A=\dfrac{201}{10100}$
