The sum of $$\frac{3}{{1 \cdot 2}} \cdot \frac{1}{2} + \frac{4}{{2 \cdot 3}} \cdot {\left( {\frac{1}{2}} \right)^2} + \frac{5}{{3 \cdot 4}} \cdot {\left( {\frac{1}{2}} \right)^3} + ......\,{\text{to }}n$$           terms is equal to

Solution :
$${t_n} = \frac{{n + 2}}{{n\left( {n + 1} \right)}} \cdot {\left( {\frac{1}{2}} \right)^n} = \frac{{2\left( {n + 1} \right) - n}}{{n\left( {n + 1} \right)}} \cdot {\left( {\frac{1}{2}} \right)^n} = \frac{1}{n} \cdot {\left( {\frac{1}{2}} \right)^{n - 1}} - \frac{1}{{n + 1}} \cdot {\left( {\frac{1}{2}} \right)^n}.$$