Polynomials, Cubic Equations and Partial Fractions - Partial Fractions

Express \(\frac{1}{\left( x+1 \right)\left( x+2 \right)}\) in partial fractions.

Hence,

1. Evaluate \(\frac{1}{\left( 2 \right)\left( 3 \right)}+\frac{1}{\left( 3 \right)\left( 4 \right)}+\frac{1}{\left( 4 \right)\left( 5 \right)}+...+\frac{1}{\left( 19 \right)\left( 20 \right)}\).
2. Find the value of \(n\) if \(\frac{1}{\left( 2 \right)\left( 3 \right)}+\frac{1}{\left( 3 \right)\left( 4 \right)}+\frac{1}{\left( 4 \right)\left( 5 \right)}+...+\frac{1}{\left( n-1 \right)\left( n \right)}=\frac{49}{100}\).