if-1-x-x-2-0-find-the-value-of-A-x-1-x-6-x-2-1-x-2-6-x-100-1-x-100-6-

Question Number 27094 by abdo imad last updated on 02/Jan/18

i f 1 + x + x^{2} = 0 f i n d t h e v a l u e o f

A = {(x + \frac{1}{x})}^{6} + {(x^{2} + \frac{1}{x^{2}})}^{6} + \dots {(x^{100} + \frac{1}{x^{100}})}^{6} .

Commented by AHSoomro last updated on 02/Jan/18

You can't use 'macro parameter character #' in math mode

Commented by abdo imad last updated on 08/Jan/18

A = \sum_{k = 1}^{100} {(x^{k} + x^{- k})}^{6} = \sum_{k = 1}^{k = 100} (x^{k} + x_{k}^{-}) b u t t h e r o o t s o f

1 + x + x^{2} = 0 a r e j = e^{i \frac{2 π}{3}} a n d j^{-} = e^{- i \frac{2 π}{3}} w e c a n c h o o s e x = e^{i \frac{2 π}{3}}

A = \sum_{k = 1}^{100} {(2 R e (j^{k}))}^{6} = 2^{6} \sum_{k = 1}^{100} c o s {(\frac{2 k π}{3})}^{6}

= 64 \sum_{k = 1}^{100} c o s {(\frac{2 k π}{3})}^{6}

b u t {(c o s x)}^{6} = {(\frac{e^{i x} + e^{- i x}}{2})}^{6} = \frac{1}{2^{6}} \sum_{k = 0}^{6} C_{6}^{k} {(e^{i x})}^{k} {(e^{- i x})}^{6 - k}

= \frac{1}{2^{6}} \sum_{k = 0}^{6} C_{6}^{k} e^{i k x} e^{- i (6 - k) x} \dots b e c o n t i n u e d \dots

Answered by bsayani309@gmail.com last updated on 05/Jan/18

1 + x + x^{2} = 0

o r . 1 + x^{2} = - x

o r . \frac{1 + x 2}{x} = - 1

o r . (\frac{1}{x} + x) = - 1

o r . {(\frac{1}{x} + x)}^{6} = 1

t h e n A = 1 + 2 n d 1 + 3 r d 1 + \dots \dots . + 100 t h 1 = 100