Let's start by determining the value of [tex]\(a\)[/tex] given [tex]\(a = \frac{3 + \sqrt{5}}{2}\)[/tex].
1. Calculate [tex]\(a\)[/tex]:
[tex]\[ a = \frac{3 + \sqrt{5}}{2} \approx 2.618033988749895 \][/tex]
2. Calculate [tex]\(a^2\)[/tex]:
Next, we need to square the value of [tex]\(a\)[/tex]:
[tex]\[ a^2 = \left(\frac{3 + \sqrt{5}}{2}\right)^2 \approx 6.854101966249685 \][/tex]
3. Calculate [tex]\(\frac{1}{a^2}\)[/tex]:
Now, we find the reciprocal of [tex]\(a^2\)[/tex]:
[tex]\[ \frac{1}{a^2} \approx \frac{1}{6.854101966249685} \approx 0.14589803375031546 \][/tex]
4. Add [tex]\(a^2\)[/tex] and [tex]\(\frac{1}{a^2}\)[/tex]:
Finally, we add these two values together:
[tex]\[ a^2 + \frac{1}{a^2} \approx 6.854101966249685 + 0.14589803375031546 = 7.0 \][/tex]
Hence, the value of [tex]\(a^2 + \frac{1}{a^2}\)[/tex] is [tex]\( \boxed{7.0} \)[/tex].
