\(\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)...\left(1-\frac{1}{225}\right)\)
\(=\frac{3}{4}\cdot\frac{8}{9}\cdot...\cdot\frac{224}{225}\)
\(=\frac{1\cdot3}{2\cdot2}\cdot\frac{2\cdot4}{3\cdot3}\cdot...\cdot\frac{14\cdot16}{15\cdot15}\)
\(=\frac{1\cdot3\cdot2\cdot4\cdot...\cdot14\cdot16}{2\cdot2\cdot3\cdot3\cdot...\cdot25\cdot25}\)
\(=\frac{\left(1\cdot2\cdot3\cdot...\cdot14\right)\cdot\left(3\cdot4\cdot5\cdot...\cdot16\right)}{\left(2\cdot3\cdot4\cdot...\cdot15\right)\cdot\left(2\cdot3\cdot4\cdot...\cdot15\right)}\)
\(=\frac{1\cdot2\cdot3\cdot...\cdot14}{2\cdot3\cdot4\cdot...\cdot15}\cdot\frac{3\cdot4\cdot5\cdot...\cdot16}{2\cdot3\cdot4\cdot...\cdot15}\)
\(=\frac{1}{15}\cdot\frac{16}{2}\)
\(=\frac{1}{15}\cdot8\)
\(=\frac{8}{15}\)
( 1- 1/4 ) . ( 1 - 1/9 ) . ( 1 - 1/16 ) ...( 1 - 1/225 )
= 3/4 . 8/9 . 15/16 ... 224/225
= 3 . 8 . 15 ... 224 / 4 . 9 . ..225
= 3 . 2 . 4 . 3 . 5 ... 14 . 16 / 2 . 2 . 3 . 3 ... 15 . 15
= ( 3 . 4 . 5 ... 16 ) . ( 2 . 3 . 4 ... 14 ) / ( 2 . 3 ... 15 ) . ( 2 . 3 ... 15 )
= 16/ 2 . 15
= `8/15
