let-and-roots-of-the-equation-x-2-x-2-0-simplify-A-p-p-p-and-calculate-p-0-n-1-A-p-and-p-0-n-1-A-p-2-

Question Number 73487 by abdomathmax last updated on 13/Nov/19

l e t α a n d β r o o t s o f t h e e q u a t i o n x^{2} - x + 2 = 0

s i m p l i f y A_{p} = α^{p} + β^{p} a n d c a l c u l a t e

\sum_{p = 0}^{n - 1} A_{p} a n d \sum_{p = 0}^{n - 1} A_{p}^{2}

Commented by mathmax by abdo last updated on 15/Nov/19

l e t s o l v e x^{2} - x + 2 = 0 \to Δ = 1 - 8 = - 7 \Rightarrow

α = \frac{1 + i \sqrt{7}}{2} a n d β = \frac{1 - i \sqrt{7}}{2}

∣ α ∣= \frac{1}{2} \sqrt{1 + 7} = \frac{2 \sqrt{2}}{2} = \sqrt{2} \Rightarrow α = \sqrt{2} e^{i a r c t a n (\sqrt{7})} a n d β = \sqrt{2} e^{- i a r c t a n (\sqrt{7})}

A_{p} = α^{p} + β^{p} = {(\sqrt{2})}^{p} e^{i p a r c t a n (\sqrt{7})} + {(\sqrt{2})}^{p} e^{- i p a r c t a n (\sqrt{7})}

= 2^{\frac{p}{2}} \times 2 R e (e^{i p a r c t a n (\sqrt{7})}) = 2^{1 + \frac{p}{2}} c o s (p a r c t a n (\sqrt{7}))

\sum_{p = 0}^{n - 1} A_{p} = \sum_{p = 0}^{n - 1} 2^{1 + \frac{p}{2}} c o s (p a r c t a n (\sqrt{7}))

= R e (2 \sum_{p = 0}^{n - 1} {(\sqrt{2})}^{p} e^{i p a r c t a n (\sqrt{7})}) w e h a v e

\sum_{p = 0}^{n - 1} {(\sqrt{2})}^{p} e^{i p a r c t a n (\sqrt{7})}) = \sum_{p = 0}^{n - 1} {(\sqrt{2} e^{i a r c t a n (\sqrt{7})})}^{p}

= \frac{1 - {(\sqrt{2} e^{i a r c t a n (\sqrt{7})})}^{n}}{1 - \sqrt{2} e^{i a r c t a n (\sqrt{7})}} = \frac{1 - {(\sqrt{2})}^{n} e^{i n a r c t a n (\sqrt{7})}}{1 - \sqrt{2} e^{i s r c t a n (\sqrt{7})}}

= \frac{(1 - {(\sqrt{2})}^{n} (c o s (n a r c t a n (\sqrt{7}) + i s i n (n a r c t a n (\sqrt{7}))}{1 - \sqrt{2} (c o s (a r c t a n (\sqrt{7}) + i s i n (a r c t a n (\sqrt{7}))}

r e s t t o e x t r a c t R e (o f t h i s q u a n t i t y \dots)