Menu Close

Given-f-x-k-0-n-n-C-k-sin-kx-cos-n-k-x-Find-a-simple-form-for-f-x-Your-answer-should-be-written-like-c-n-g-nx-




Question Number 50908 by Smail last updated on 22/Dec/18
Given f(x)=Σ_(k=0) ^n ^n C_k sin(kx)cos((n−k)x)  Find a simple form for f(x)  (Your answer should be written like c(n).g(nx))
$${Given}\:{f}\left({x}\right)=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{k}} {sin}\left({kx}\right){cos}\left(\left({n}−{k}\right){x}\right) \\ $$$${Find}\:{a}\:{simple}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left({Your}\:{answer}\:{should}\:{be}\:{written}\:{like}\:{c}\left({n}\right).{g}\left({nx}\right)\right)\: \\ $$
Answered by Smail last updated on 23/Dec/18
f(x)=Σ_(k=0) ^n ^n C_k sin(kx)cos((n−k)x)  let l=n−k  f(x)=Σ_(l=0) ^n ^n C_(n−l) sin((n−l)x)cos(lx)  =Σ_(l=0) ^n ^n C_l sin((n−l)x)cos(lx)  2f(x)=Σ_(k=0) ^n ^n C_k (sin((n−k)x)cos(kx)+sin(kx)cos((n−k)x))  =Σ_(k=0) ^n ^n C_k (sin((n−k)x+kx))  2f(x)=Σ_(k=0) ^n ^n C_k sin(nx)=2^n sin(nx)  f(x)=2^(n−1) sin(nx)
$${f}\left({x}\right)=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{k}} {sin}\left({kx}\right){cos}\left(\left({n}−{k}\right){x}\right) \\ $$$${let}\:{l}={n}−{k} \\ $$$${f}\left({x}\right)=\underset{{l}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{n}−{l}} {sin}\left(\left({n}−{l}\right){x}\right){cos}\left({lx}\right) \\ $$$$=\underset{{l}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{l}} {sin}\left(\left({n}−{l}\right){x}\right){cos}\left({lx}\right) \\ $$$$\mathrm{2}{f}\left({x}\right)=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{k}} \left({sin}\left(\left({n}−{k}\right){x}\right){cos}\left({kx}\right)+{sin}\left({kx}\right){cos}\left(\left({n}−{k}\right){x}\right)\right) \\ $$$$=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{k}} \left({sin}\left(\left({n}−{k}\right){x}+{kx}\right)\right) \\ $$$$\mathrm{2}{f}\left({x}\right)=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\:^{{n}} {C}_{{k}} {sin}\left({nx}\right)=\mathrm{2}^{{n}} {sin}\left({nx}\right) \\ $$$${f}\left({x}\right)=\mathrm{2}^{{n}−\mathrm{1}} {sin}\left({nx}\right) \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *