\(S=\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right).....\left(\frac{1}{100^2}-1\right)\)
\(S=\left(\frac{1}{4}-1\right)\left(\frac{1}{9}-1\right)......\left(\frac{1}{10000}-1\right)\)
\(S=-\frac{3}{4}.\left(-\frac{8}{9}\right).....\left(-\frac{9999}{10000}\right)\)
\(S=\frac{1.3.2.4.....99.101}{2.2.3.3....100.100}\)
\(S=\frac{\left(1.2.3.....99\right).\left(3.4.5....101\right)}{\left(2.3....100\right).\left(2.3.....100\right)}\)
\(S=\frac{1.101}{100.2}\)
\(S=\frac{101}{200}\)