\(\left(\frac{1}{2^2}-1\right).\left(\frac{1}{3^2}-1\right)\left(\frac{1}{4^2}-1\right)...\left(\frac{1}{50^2}-1\right)\)
\(=-\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)...\left(1-\frac{1}{50^2}\right)\)
\(=-\frac{2^2-1}{2^2}.\frac{3^2-1}{3^2}.\frac{4^2-1}{4^2}....\frac{50^2-1}{50^2}\)
\(=-\frac{\left(2-1\right)\left(2+1\right)}{2^2}.\frac{\left(3-1\right)\left(3+1\right)}{3^2}.\frac{\left(4-1\right)\left(4+1\right)}{4^2}...\frac{\left(50-1\right)\left(50+1\right)}{50^2}\)
\(=-\frac{1.3}{2^2}.\frac{2.4}{3^2}.\frac{3.5}{4^2}...\frac{49.51}{50}\)
\(=-\frac{1.2.3...49}{2.3.4...50}.\frac{3.4.5...51}{2.3.4...50}\)
\(=-\frac{1}{50}.\frac{51}{2}=-\frac{51}{100}\)
