Find the value
\[ a = \frac{3 + \sqrt{5}}{2} \]
\[ \frac{1}{a} = \frac{2}{3 + \sqrt{5}} \times \frac{3 – \sqrt{5}}{3 – \sqrt{5}} = \frac{3 – \sqrt{5}}{2} \]
\[ a + \frac{1}{a} = \frac{3 + \sqrt{5}}{2} + \frac{3 – \sqrt{5}}{2} = 3 \]
\[ a^2 + \frac{1}{a^2} = (a + \frac{1}{a})^2 – 2 \]
\[ = 3^2 – 2 = 9 – 2 = 7 \]