find-the-sequence-U-n-wich-verify-U-n-U-n-1-sin-n-n-from-n- Tinku Tara June 3, 2023 Relation and Functions 0 Comments FacebookTweetPin Question Number 67023 by mathmax by abdo last updated on 21/Aug/19 findthesequenceUnwichverifyUn+Un+1=sin(n)∀nfromn Commented by mathmax by abdo last updated on 23/Aug/19 wehaveun+un+1=sin(n)⇒∑k=0n−1(uk+uk+1)(−1)k=∑k=0n(−1)ksin(k)⇒u0+u1−u1−u2+u2+u3+….+(−1)n−1(un−1+un)=∑k=0n(−1)ksin(k)⇒u0+(−1)n−1un=∑k=0n(−1)ksin(k)⇒u0−∑k=0n(−1)ksin(k)=(−1)nun⇒un=(−1)nu0−(−1)n∑k=0n(−1)ksin(k)letfindwn=∑k=0n(−1)ksin)k)wn=Im(∑k=0n(−1)keik)=Im(∑k=0n(−ei)k)and∑k=0n(−ei)k=1−(−ei)n+11+ei=1+(−1)nei(n+1)1+ei=1+(−1)nei(n+1)1+cos(1)+isin(1)=(1+cos(1)−isin(1))(1+(−1)nei(n+1))(1+cos(1))2+sin21=(1+cos(1)−isin(1))(1+(−1)ncos(n+1)+i(−1)nsin(n+1))(1+cos(1))2+sin21=(1+cos(1))(1+(−1)ncos(n+1))+i(−1)nsin(n+1)(1+cos(1))−isin(1)(1+(−1)ncos(n+1))+….(1+cos(1))2+sin2(1)⇒wn=(−1)nsin(n+1)(1+cos(1))−sin(1)(1+(−1)ncos)n+1))2+2cos(1)⇒un=(−1)nu0−(−1)nwn Answered by mind is power last updated on 23/Aug/19 letvn+1+vn=cos(n)andZn=vn+iunzn+1+zn=cos(n)+isin(n)=ein==>zn+1ei(n)+znein=1=eizn+1ei(n+1)+Znein=1letWn=znein===>eiWn+1+Wn=1eix+1=0==>x=−1eix=−e−ixWn=c(−1)ne−inx+tnwithetnparticularsolutionofeitn+1+tn=1lettn=an+b==>ei(an+a+b)+an+b=1∀n∈IN==>an(ei+1)=0andb(ei+1)=1==>a=0andb=11+ei==>tn=11+eiWn=(−1)nce−in+11+ei.c∈IC.==>Zn=einWn=c(−1)n+ein1+ei=c(−1)n+ein+ei(n−1)2+2cos(1)un=IM(Zn)=k(−1)n+12+2cos(1)[sin(n)+sin(n−1)]sin(a)+sin(b)=2cos(a−b2)sin(a+b2).and2+2cos(1)=2+2cos(2.12)=4cos2(12)wefindun=k(−1)n+2cos(12)sin(n−12)4cos2(12)=k(−1)n+sin(n−12)2cos(12) Commented by mathmax by abdo last updated on 23/Aug/19 thankyousir. Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: While-this-may-be-a-bit-off-the-usual-Q-n-A-posts-here-what-advice-can-be-given-to-someone-wishing-to-successfully-carry-out-undergraduate-studies-in-mathematics-I-m-worried-about-my-problem-solvNext Next post: Question-67025 Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.