Jippie:-)

\(\begin{array}{l} {\left( {1\frac{1}{2}} \right)^2} = {\left( {1 + \frac{1}{2}} \right)^2} = {1^2} + 2 \cdot 1 \cdot \frac{1}{2} + {\left( {\frac{1}{2}} \right)^2} = 1 + 1 + \frac{1}{4} = 2\frac{1}{4}\\ {\left( {2\frac{1}{2}} \right)^2} = {\left( {2 + \frac{1}{2}} \right)^2} = {2^2} + 2 \cdot 2 \cdot \frac{1}{2} + {\left( {\frac{1}{2}} \right)^2} = 4 + 2 + \frac{1}{4} = 6\frac{1}{4}\\ {\left( {3\frac{1}{2}} \right)^2} = {\left( {3 + \frac{1}{2}} \right)^2} = {3^2} + 2 \cdot 3 \cdot \frac{1}{2} + {\left( {\frac{1}{2}} \right)^2} = 9 + 3 + \frac{1}{4} = 12\frac{1}{4}\\ ...\\ {\left( {n + \frac{1}{2}} \right)^2} = {n^2} + 2 \cdot n \cdot \frac{1}{2} + {\left( {\frac{1}{2}} \right)^2} = {n^2} + n + \frac{1}{4} \end{array}\)