1-5-2-15-a-b-5-2-find-a-b-

Question Number 182627 by manxsol last updated on 12/Dec/22

{(\frac{1 + \sqrt{5}}{2})}^{15} = \frac{a + b \sqrt{5}}{2}

f i n d a + b

Answered by Rasheed.Sindhi last updated on 12/Dec/22

{(\frac{1 + \sqrt{5}}{2})}^{15} = \frac{a + b \sqrt{5}}{2}; a + b = ?

∙ {(\frac{1 + \sqrt{5}}{2})}^{3} = \frac{1 + 5 \sqrt{5} + 3 \sqrt{5} (1 + \sqrt{5})}{2^{3}}

= \frac{1 + 8 \sqrt{5} + 15}{8} = 2 + \sqrt{5}

∙ {{(\frac{1 + \sqrt{5}}{2})}^{3}}^{2} = {(2 + \sqrt{5})}^{2} = 4 + 5 + 4 \sqrt{5} = 9 + 4 \sqrt{5}

∙ {({{(\frac{1 + \sqrt{5}}{2})}^{3}}^{2})}^{2} = {(9 + 4 \sqrt{5})}^{2} = 81 + 80 + 72 \sqrt{5} = 161 + 72 \sqrt{5}

∙ {({{(\frac{1 + \sqrt{5}}{2})}^{3}}^{2})}^{2} {(\frac{1 + \sqrt{5}}{2})}^{3}

{(\frac{1 + \sqrt{5}}{2})}^{15} = (161 + 72 \sqrt{5}) (2 + \sqrt{5})

= 322 + 144 \sqrt{5} + 161 \sqrt{5} + 360

= 682 + 305 \sqrt{5}

= \frac{2 (682 + 305 \sqrt{5})}{2}

= \frac{1364 + 610 \sqrt{5}}{2} = \frac{a + b \sqrt{5}}{2}

a + b = 1364 + 610 = 1974

Commented by manxsol last updated on 12/Dec/22

t h a n k s, S i r . R a s h e e d f o r w o r k

. i t i s m y g i f t

{(\frac{1 + \sqrt{5}}{2})}^{15} = \frac{1364 + 610 \sqrt{5}}{2}

s e r i e d e f i n o n a c c i

\begin{array}{cc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 55 & 89 & 144 & 233 & 377 & 610 & 987 \end{array}

a = 610

b = 377 + 987 = 1364

d e d u c c i o n h e c h a d e t a b l a d e

p o t e n c i a s d e Φ = (1 + \sqrt{5}) / 2

Commented by Rasheed.Sindhi last updated on 12/Dec/22

A nice gift!

T han k s a lOt sir manxsol!