Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

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

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) )

$${Given}\:{X}=\left(\sqrt{\sqrt{{x}}}+\frac{\mathrm{1}}{\:\sqrt{\sqrt{{x}}}}\right)^{{n}} . \\ $$$${What}\:{is}\:{the}\:{coefficient}\:{of}\:{x}^{\mathrm{5}−\frac{\mathrm{n}}{\mathrm{4}}\:?} \\ $$$$\mathrm{propositions}: \\ $$$${a}.\:\:\:\:\mathrm{5}\frac{{n}!}{\mathrm{10}!} \\ $$$${b}.\:\:\:\:\frac{{n}!}{\mathrm{5}!} \\ $$$${c}.\:\:\:\:\:\begin{pmatrix}{{n}}\\{\mathrm{10}}\end{pmatrix} \\ $$$${d}.\:\:\:\:\:\:\begin{pmatrix}{{n}}\\{\mathrm{5}}\end{pmatrix} \\ $$

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

X=((√(√x))+(1/( (√(√x)))))^n , let x=u^4 . Find coef(u^(20−n) ) =(u+(1/u))^n =Σ_(r=0) ^n ^n C_r u^r ((1/u))^(n−r) =Σ_(r=0) ^n ^n C_r u^(2r−n) r=10 ⇒coef(u^(20−n) )= ^n C_(10) = ((n),((10)) )

$${X}=\left(\sqrt{\sqrt{\mathrm{x}}}+\frac{\mathrm{1}}{\:\sqrt{\sqrt{\mathrm{x}}}}\right)^{\mathrm{n}} ,\:\mathrm{let}\:\mathrm{x}=\mathrm{u}^{\mathrm{4}} .\:\mathrm{Find}\:\mathrm{coef}\left(\mathrm{u}^{\mathrm{20}−\mathrm{n}} \right) \\ $$$$\:\:\:\:=\left(\mathrm{u}+\frac{\mathrm{1}}{\mathrm{u}}\right)^{\mathrm{n}} =\underset{\mathrm{r}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{r}} \mathrm{u}^{\mathrm{r}} \left(\frac{\mathrm{1}}{\mathrm{u}}\right)^{\mathrm{n}−\mathrm{r}} =\underset{\mathrm{r}=\mathrm{0}} {\overset{\mathrm{n}} {\sum}}\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{r}} \mathrm{u}^{\mathrm{2r}−\mathrm{n}} \\ $$$$\mathrm{r}=\mathrm{10}\:\Rightarrow\mathrm{coef}\left(\mathrm{u}^{\mathrm{20}−\mathrm{n}} \right)=\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{10}} =\begin{pmatrix}{\mathrm{n}}\\{\mathrm{10}}\end{pmatrix} \\ $$

Answered by iloveisrael last updated on 15/May/21

X= Σ_(k=0) ^n ((n),(k) )(x^(1/4) )^k (x^(−(1/4)) )^(n−k) ⇒x^((k+k−n)/4) = x^(5−(n/4)) ; ((2k−n)/4) = ((20−n)/4) k=10 ⇒ coefficient = ((( n)),((10)) )

$${X}=\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}\left({x}^{\frac{\mathrm{1}}{\mathrm{4}}} \right)^{{k}} \left({x}^{−\frac{\mathrm{1}}{\mathrm{4}}} \right)^{{n}−{k}} \\ $$$$\Rightarrow{x}^{\frac{{k}+{k}−{n}}{\mathrm{4}}} =\:{x}^{\mathrm{5}−\frac{{n}}{\mathrm{4}}} \:;\:\frac{\mathrm{2}{k}−{n}}{\mathrm{4}}\:=\:\frac{\mathrm{20}−{n}}{\mathrm{4}} \\ $$$${k}=\mathrm{10}\:\Rightarrow\:\mathrm{coefficient}\:=\:\begin{pmatrix}{\:{n}}\\{\mathrm{10}}\end{pmatrix} \\ $$

Answered by mathocean1 last updated on 15/May/21

thank you sirs

$${thank}\:{you}\:{sirs} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com