If-x-1-5-2-find-x-12-

Question Number 203264 by hardmath last updated on 13/Jan/24

If x = \frac{1 + \sqrt{5}}{2} find : x^{12} = ?

Answered by MM42 last updated on 13/Jan/24

{\begin{cases} x^{2} = \frac{3 + \sqrt{5}}{2} \\ x^{4} = \frac{7 + 3 \sqrt{5}}{2} \end{cases} \Rightarrow x^{6} = 9 + 4 \sqrt{5}

\Rightarrow x^{12} = 161 + 72 \sqrt{5}

Commented by hardmath last updated on 13/Jan/24

▴ x = \frac{1 + \sqrt{5}}{2} = φ

▴ φ^{n} = F_{n} φ + F_{n - 1}

φ^{12} = F_{12} φ + F_{11}

= 144 φ + 89

= 144 (\frac{1 + \sqrt{5}}{2}) + 89

= 161 + 72 \sqrt{5} ✓

Answered by Rasheed.Sindhi last updated on 13/Jan/24

{(2 x - 1)}^{2} = 5

4 x^{2} - 4 x - 4 = 0

x^{2} = x + 1 \Rightarrow x^{3} = x^{2} + x = 2 x + 1

\Rightarrow x^{4} = 2 x^{2} + x = 3 x + 2

\Rightarrow x^{5} = 3 x^{2} + 2 x = 5 x + 3

\Rightarrow x^{6} = 5 x^{2} + 3 x = 8 x + 5

\Rightarrow x^{12} = 64 x^{2} + 80 x + 25 = 144 x + 89

= 144 (\frac{1 + \sqrt{5}}{2}) + 89

= 72 + 72 \sqrt{5} + 89

= 161 + 72 \sqrt{5}