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Question Number 65297 by mathmax by abdo last updated on 28/Jul/19
let U_n a sequence wich verify U_n +U_(n+1) +U_(n+2) =n(−1)^n for all integr n calculate interms of n A_n =Σ_(k=0) ^n (−1)^k U_k the first term is U_0
$${let}\:\:\:{U}_{{n}} \:\:{a}\:{sequence}\:{wich}\:{verify}\:\:{U}_{{n}} \:+{U}_{{n}+\mathrm{1}} +{U}_{{n}+\mathrm{2}} \:={n}\left(−\mathrm{1}\right)^{{n}} \\ $$$${for}\:{all}\:{integr}\:{n}\:\:\:{calculate}\:{interms}\:{of}\:{n} \\ $$$${A}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\left(−\mathrm{1}\right)^{{k}} \:{U}_{{k}} \\ $$$${the}\:{first}\:{term}\:{is}\:{U}_{\mathrm{0}} \\ $$
Commented by mathmax by abdo last updated on 29/Jul/19
we have u_n +u_(n+1) +u_(n+2) =n(−1)^n ⇒ Σ_(k=0) ^n (−1)^k (u_k +u_(k+1) )+Σ_(k=0) ^n (−1)^k u_(k+2) =Σ_(k=0) ^n k(−1)^k Σ_(k=0) ^n (−1)^k (u_k +u_(k+1) ) =u_0 +u_1 −u_1 −u_2 +....(−1)^(n−1) (u_(n−1) +u_n ) +(−1)^n (u_n +u_(n+1) ) =u_0 +(−1)^n u_(n+1) ⇒ Σ_(k=0) ^n (−1)^k u_(k+2) =Σ_(k=0) ^n k(−1)^k −u_0 −(−1)^n u_(n+1) Σ_(k=2) ^(n+2) (−1)^(k−2) u_k =Σ_(k=0) ^n k(−1)^k −u_0 −(−1)^n u_(n+1) ⇒ Σ_(k=0) ^(n+2) (−1)^k u_k −u_0 +u_1 =Σ_(k=0) ^n k(−1)^k −u_0 −(−1)^n u_(n+1) ⇒ Σ_(k=0) ^n (−1)^k u_k +(−1)^(n+1) u_(n+1) +(−1)^(n+2) u_(n+2) = Σ_(k=0) ^n k(−1)^k −(−1)^n u_(n+1) −u_1 ⇒ A_n =Σ_(k=0) ^n k(−1)^k −(−1)^n u_(n+1) +(−1)^n u_(n+1) −(−1)^n u_(n+2) −u_1 =Σ_(k=0) ^n k(−1)^k −(−1)^n u_(n+2) −u_1 let p(x) =Σ_(k=0) ^n x^k ⇒p^′ (x) =Σ_(k=1) ^n kx^(k−1) ⇒xp^′ (x) =Σ_(k=1) ^n kx^k but p(x) =((x^(n+1) −1)/(x−1)) (x≠1) ⇒p^′ (x) =(((n+1)x^n (x−1)−(x^(n+1) −1))/((x−1)^2 )) =((nx^(n+1) −(n+1)x^n +1)/((x−1)^2 )) ⇒Σ_(k=0) ^n kx^k =((nx^(n+2) −(n+1)x^(n+1) +x)/((x−1)^2 )) x=−1 ⇒Σ_(k=0) ^n k(−1)^k =((n(−1)^(n+2) −(n+1)(−1)^(n+1) −1)/4) =((n(−1)^n +(n+1)(−1)^n −1)/4) =(((2n+1)(−1)^n −1)/4) ⇒ A_n =(((2n+1)(−1)^n −1)/4) −(−1)^n u_(n+2) −u_1
$${we}\:{have}\:{u}_{{n}} \:+{u}_{{n}+\mathrm{1}} \:+{u}_{{n}+\mathrm{2}} ={n}\left(−\mathrm{1}\right)^{{n}} \:\Rightarrow \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}} \left(−\mathrm{1}\right)^{{k}} \:\left({u}_{{k}} \:+{u}_{{k}+\mathrm{1}} \right)+\sum_{{k}=\mathrm{0}} ^{{n}} \left(−\mathrm{1}\right)^{{k}} \:{u}_{{k}+\mathrm{2}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}\left(−\mathrm{1}\right)^{{k}} \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}} \left(−\mathrm{1}\right)^{{k}} \left({u}_{{k}} \:+{u}_{{k}+\mathrm{1}} \right)\:={u}_{\mathrm{0}} +{u}_{\mathrm{1}} −{u}_{\mathrm{1}} −{u}_{\mathrm{2}} \:+….\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} \left({u}_{{n}−\mathrm{1}} \:+{u}_{{n}} \right) \\ $$$$+\left(−\mathrm{1}\right)^{{n}} \left({u}_{{n}} \:+{u}_{{n}+\mathrm{1}} \right)\:={u}_{\mathrm{0}} \:+\left(−\mathrm{1}\right)^{{n}} {u}_{{n}+\mathrm{1}} \Rightarrow \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}} \left(−\mathrm{1}\right)^{{k}} {u}_{{k}+\mathrm{2}} =\sum_{{k}=\mathrm{0}} ^{{n}} {k}\left(−\mathrm{1}\right)^{{k}} −{u}_{\mathrm{0}} −\left(−\mathrm{1}\right)^{{n}} {u}_{{n}+\mathrm{1}} \\ $$$$\sum_{{k}=\mathrm{2}} ^{{n}+\mathrm{2}} \left(−\mathrm{1}\right)^{{k}−\mathrm{2}} \:{u}_{{k}} =\sum_{{k}=\mathrm{0}} ^{{n}} {k}\left(−\mathrm{1}\right)^{{k}} −{u}_{\mathrm{0}} −\left(−\mathrm{1}\right)^{{n}} {u}_{{n}+\mathrm{1}} \Rightarrow \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}+\mathrm{2}} \left(−\mathrm{1}\right)^{{k}} {u}_{{k}} −{u}_{\mathrm{0}} +{u}_{\mathrm{1}} =\sum_{{k}=\mathrm{0}} ^{{n}} {k}\left(−\mathrm{1}\right)^{{k}} −{u}_{\mathrm{0}} −\left(−\mathrm{1}\right)^{{n}} {u}_{{n}+\mathrm{1}} \:\Rightarrow \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}} \:\left(−\mathrm{1}\right)^{{k}} \:{u}_{{k}} \:\:+\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} {u}_{{n}+\mathrm{1}} +\left(−\mathrm{1}\right)^{{n}+\mathrm{2}} {u}_{{n}+\mathrm{2}} = \\ $$$$\sum_{{k}=\mathrm{0}} ^{{n}} {k}\left(−\mathrm{1}\right)^{{k}} −\left(−\mathrm{1}\right)^{{n}} {u}_{{n}+\mathrm{1}} −{u}_{\mathrm{1}} \Rightarrow \\ $$$${A}_{{n}} =\sum_{{k}=\mathrm{0}} ^{{n}} {k}\left(−\mathrm{1}\right)^{{k}} −\left(−\mathrm{1}\right)^{{n}} {u}_{{n}+\mathrm{1}} +\left(−\mathrm{1}\right)^{{n}} {u}_{{n}+\mathrm{1}} −\left(−\mathrm{1}\right)^{{n}} {u}_{{n}+\mathrm{2}} \:−{u}_{\mathrm{1}} \\ $$$$=\sum_{{k}=\mathrm{0}} ^{{n}} {k}\left(−\mathrm{1}\right)^{{k}} \:\:−\left(−\mathrm{1}\right)^{{n}} {u}_{{n}+\mathrm{2}} −{u}_{\mathrm{1}} \\ $$$${let}\:{p}\left({x}\right)\:=\sum_{{k}=\mathrm{0}} ^{{n}} \:{x}^{{k}} \:\Rightarrow{p}^{'} \left({x}\right)\:=\sum_{{k}=\mathrm{1}} ^{{n}} {kx}^{{k}−\mathrm{1}} \:\Rightarrow{xp}^{'} \left({x}\right)\:=\sum_{{k}=\mathrm{1}} ^{{n}} {kx}^{{k}} \\ $$$${but}\:{p}\left({x}\right)\:=\frac{{x}^{{n}+\mathrm{1}} −\mathrm{1}}{{x}−\mathrm{1}}\:\left({x}\neq\mathrm{1}\right)\:\Rightarrow{p}^{'} \left({x}\right)\:=\frac{\left({n}+\mathrm{1}\right){x}^{{n}} \left({x}−\mathrm{1}\right)−\left({x}^{{n}+\mathrm{1}} −\mathrm{1}\right)}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$=\frac{{nx}^{{n}+\mathrm{1}} −\left({n}+\mathrm{1}\right){x}^{{n}} \:+\mathrm{1}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} }\:\Rightarrow\sum_{{k}=\mathrm{0}} ^{{n}} {kx}^{{k}} \:=\frac{{nx}^{{n}+\mathrm{2}} −\left({n}+\mathrm{1}\right){x}^{{n}+\mathrm{1}} \:+{x}}{\left({x}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$${x}=−\mathrm{1}\:\Rightarrow\sum_{{k}=\mathrm{0}} ^{{n}} \:{k}\left(−\mathrm{1}\right)^{{k}} \:=\frac{{n}\left(−\mathrm{1}\right)^{{n}+\mathrm{2}} −\left({n}+\mathrm{1}\right)\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} −\mathrm{1}}{\mathrm{4}} \\ $$$$=\frac{{n}\left(−\mathrm{1}\right)^{{n}} +\left({n}+\mathrm{1}\right)\left(−\mathrm{1}\right)^{{n}} −\mathrm{1}}{\mathrm{4}}\:=\frac{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(−\mathrm{1}\right)^{{n}} −\mathrm{1}}{\mathrm{4}}\:\Rightarrow \\ $$$${A}_{{n}} =\frac{\left(\mathrm{2}{n}+\mathrm{1}\right)\left(−\mathrm{1}\right)^{{n}} −\mathrm{1}}{\mathrm{4}}\:−\left(−\mathrm{1}\right)^{{n}} {u}_{{n}+\mathrm{2}} \:−{u}_{\mathrm{1}} \\ $$$$ \\ $$$$ \\ $$

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