Given-X-x-1-x-n-What-is-the-coefficient-of-x-5-n-4-propositions-a-5-n-10-b-n-5-c-n-10-d-n-5-

Question Number 141024 by mathocean1 last updated on 15/May/21

G i v e n X = {(\sqrt{\sqrt{x}} + \frac{1}{\sqrt{\sqrt{x}}})}^{n} .

W h a t i s t h e c o e f f i c i e n t o f x^{5 - \frac{n}{4} ?}

propositions :

a . 5 \frac{n!}{10!}

b . \frac{n!}{5!}

c . (\begin{matrix} n \\ 10 \end{matrix})

d . (\begin{matrix} n \\ 5 \end{matrix})

Answered by Ar Brandon last updated on 15/May/21

X = {(\sqrt{\sqrt{x}} + \frac{1}{\sqrt{\sqrt{x}}})}^{n}, let x = u^{4} . Find coef (u^{20 - n})

= {(u + \frac{1}{u})}^{n} = \underset{r = 0}{\sum^{n}} \overset{n}{} C_{r} u^{r} {(\frac{1}{u})}^{n - r} = \underset{r = 0}{\sum^{n}} \overset{n}{} C_{r} u^{2 r - n}

r = 10 \Rightarrow coef (u^{20 - n}) = \overset{n}{} C_{10} = (\begin{matrix} n \\ 10 \end{matrix})

Answered by iloveisrael last updated on 15/May/21

X = \underset{k = 0}{\sum^{n}} (\begin{matrix} n \\ k \end{matrix}) {(x^{\frac{1}{4}})}^{k} {(x^{- \frac{1}{4}})}^{n - k}

\Rightarrow x^{\frac{k + k - n}{4}} = x^{5 - \frac{n}{4}}; \frac{2 k - n}{4} = \frac{20 - n}{4}

k = 10 \Rightarrow coefficient = (\begin{matrix} n \\ 10 \end{matrix})

Answered by mathocean1 last updated on 15/May/21

t h a n k y o u s i r s