If-x-1-x-1-find-x-61-1-x-61-4-

Question Number 214186 by hardmath last updated on 30/Nov/24

If x + \frac{1}{x} = 1 find x^{61} + \frac{1}{x^{61}} + 4 = ?

Answered by Rasheed.Sindhi last updated on 30/Nov/24

x + \frac{1}{x} = 1; x^{61} + \frac{1}{x^{61}} + 4 = ?

x + \frac{1}{x} = 1 \Rightarrow x^{2} = x - 1

x^{3} = x^{2} - x = (x - 1) - x = - 1

{(x^{3})}^{20} = {(- 1)}^{20} = 1 \Rightarrow x^{60} = 1

x^{61} + \frac{1}{x^{61}} + 4 = x^{60} . x + \frac{1}{x^{60} . x} + 4

= x + \frac{1}{x} + 4 = 1 + 4 = 5

Answered by BaliramKumar last updated on 01/Dec/24

x + \frac{1}{x} = 2 c o s (\frac{π}{3})

x^{61} + \frac{1}{x^{61}} = 2 c o s (\frac{61 π}{3}) x^{n} + \frac{1}{x^{n}} = 2 c o s (n θ)

x^{61} + \frac{1}{x^{61}} = 2 c o s (\frac{60 π + π}{3}) \Rightarrow 2 c o s (20 π + \frac{π}{3})

x^{61} + \frac{1}{x^{61}} = 2 c o s (\frac{π}{3})

x^{61} + \frac{1}{x^{61}} = 1

x^{61} + \frac{1}{x^{61}} + 4 = 1 + 4 = 5