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Question-81308




Question Number 81308 by naka3546 last updated on 11/Feb/20
Commented by mr W last updated on 11/Feb/20
2104?
$$\mathrm{2104}? \\ $$
Commented by naka3546 last updated on 11/Feb/20
please, show  your  working, sir.
$${please},\:{show}\:\:{your}\:\:{working},\:{sir}. \\ $$
Answered by mr W last updated on 12/Feb/20
given is the recursive relation  x_(n+13) =x_(n+4) +2x_n   we see x_n =Aq^n  fulfills this condition.  Aq^(n+13) =Aq^(n+4) +2Aq^n   ⇒q^(13) =q^4 +2  it has 13 roots, say q_r  (r=1,2,...,13).  the general formula for term x_n  is then  x_n =Σ_(r=1) ^(13) A_r q_r ^n   with q_r ^(13) =q_r ^4 +2   (r=1,2,3,...,13)    x_0 =Σ_(r=1) ^(13) A_r =1  (x_(13) =x_4 +2x_0  ⇒x_0 =1)  x_m =Σ_(r=1) ^(13) A_r q_r ^m =0   (m=1,2,3,...,12)  x_(13) =Σ_(r=1) ^(13) A_r q_r ^(13) =2    x_(143) =Σ_(r=1) ^(13) A_r q_r ^(143)   q_r ^(143) =(q_r ^(13) )^(11) =(q_r ^4 +2)^(11) =Σ_(k=0) ^(11) C_k ^(11) q_r ^(4k) 2^(11−k)     x_(143) =Σ_(r=1) ^(13) A_r (Σ_(k=0) ^(11) C_k ^(11) q_r ^(4k) 2^(11−k) )  x_(143) =Σ_(k=0) ^(11) C_k ^(11) 2^(11−k) (Σ_(r=1) ^(13) A_r q_r ^(4k) )  k=0 ⇒ Σ_(r=1) ^(13) A_r q_r ^(4k) =1  k=1..3 ⇒ Σ_(r=1) ^(13) A_r q_r ^(4k) =0  ⇒x_(134) =2^(11) +Σ_(k=4) ^(11) C_k ^(11) 2^(11−k) (Σ_(r=1) ^(13) A_r q_r ^(4k) )  k=4..11:  q_r ^(16) =q_r ^(13) q_r ^3 =(q_r ^4 +2)q_r ^3 =q_r ^7 +2q_r ^3   ⇒Σ_(r=1) ^(13) A_r q_r ^(16) =Σ_(r=1) ^(13) A_r q_r ^7 +2Σ_(r=1) ^(13) A_r q_r ^3 =0  q_r ^(20) =q_r ^(13) q_r ^7 =(q_r ^4 +2)q_r ^7 =q_r ^(11) +2q_r ^7   ⇒Σ_(r=1) ^(13) A_r q_r ^(20) =Σ_(r=1) ^(13) A_r q_r ^(11) +2Σ_(r=1) ^(13) A_r q_r ^7 =0  q_r ^(24) =q_r ^(13) q_r ^(11) =(q_r ^4 +2)q_r ^(11) =(q_r ^4 +2)q_r ^2 +2q_r ^(11)   ⇒Σ_(r=1) ^(13) A_r q_r ^(24) =0  q_r ^(28) =(q_r ^(13) )^2 q_r ^2 =(q_r ^4 +2)^2 q_r ^2 =q_r ^(10) +4q_r ^6 +4q_r ^2   ⇒Σ_(r=1) ^(13) A_r q_r ^(28) =0  q_r ^(32) =(q_r ^(13) )^2 q_r ^6 =(q_r ^4 +2)^2 q_r ^6 =q_r ^(14) +4q_r ^(10) +4q_r ^6   =(q_r ^4 +2)q_r +4q_r ^(10) +4q_r ^6   ⇒Σ_(r=1) ^(13) A_r q_r ^(32) =0  q_r ^(36) =(q_r ^(13) )^2 q_r ^(10) =(q_r ^4 +2)^2 q_r ^(10) =(q_r ^8 +4q_r ^4 +4)q_r ^(10)   =q_r ^(18) +4q_r ^(14) +4q_r ^(10)   =(q_r ^4 +2)q_r ^5 +4(q_r ^4 +2)q_r +4q_r ^(10)   ⇒Σ_(r=1) ^(13) A_r q_r ^(36) =0  q_r ^(40) =(q_r ^(13) )^3 q_r =(q_r ^4 +2)^3 q_r =(q_r ^(12) +6q_r ^8 +12q_r ^4 +8)q_r   =q_r ^(13) +6q_r ^9 +12q_r ^5 +8q_r    ⇒Σ_(r=1) ^(13) A_r q_r ^(40) =Σ_(r=1) ^(13) A_r q_r ^(13) =2  q_r ^(44) =(q_r ^(13) )^3 q_r ^5 =(q_r ^4 +2)^3 q_r ^5 =(q_r ^(12) +6q_r ^8 +12q_r ^4 +8)q_r ^5    =q_r ^(17) +6q_r ^(13) +12q_r ^9 +8q_r ^5   =(q_r ^4 +2)q_r ^4 +6q_r ^(13) +12q_r ^9 +8q_r ^5   ⇒Σ_(r=1) ^(13) A_r q_r ^(44) =6Σ_(r=1) ^(13) A_r q_r ^(13) =6×2=12  x_(143) =2^(11) +C_(10) ^(11) 2^1 (Σ_(r=1) ^(13) A_r q_r ^(40) )+C_(11) ^(11) 2^0 (Σ_(r=1) ^(13) A_r q_r ^(44) )  x_(143) =2^(11) +11×2×2+1×2^0 ×12  ⇒x_(143) =2^(11) +44+12=2104    in this way we can calculate every  term of the sequence even though  we don′t know the values of A_r  and  q_r  (r=1,2,...,13).
$${given}\:{is}\:{the}\:{recursive}\:{relation} \\ $$$${x}_{{n}+\mathrm{13}} ={x}_{{n}+\mathrm{4}} +\mathrm{2}{x}_{{n}} \\ $$$${we}\:{see}\:{x}_{{n}} ={Aq}^{{n}} \:{fulfills}\:{this}\:{condition}. \\ $$$${Aq}^{{n}+\mathrm{13}} ={Aq}^{{n}+\mathrm{4}} +\mathrm{2}{Aq}^{{n}} \\ $$$$\Rightarrow{q}^{\mathrm{13}} ={q}^{\mathrm{4}} +\mathrm{2} \\ $$$${it}\:{has}\:\mathrm{13}\:{roots},\:{say}\:{q}_{{r}} \:\left({r}=\mathrm{1},\mathrm{2},…,\mathrm{13}\right). \\ $$$${the}\:{general}\:{formula}\:{for}\:{term}\:{x}_{{n}} \:{is}\:{then} \\ $$$${x}_{{n}} =\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{{n}} \\ $$$${with}\:{q}_{{r}} ^{\mathrm{13}} ={q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\:\:\:\left({r}=\mathrm{1},\mathrm{2},\mathrm{3},…,\mathrm{13}\right) \\ $$$$ \\ $$$${x}_{\mathrm{0}} =\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} =\mathrm{1}\:\:\left({x}_{\mathrm{13}} ={x}_{\mathrm{4}} +\mathrm{2}{x}_{\mathrm{0}} \:\Rightarrow{x}_{\mathrm{0}} =\mathrm{1}\right) \\ $$$${x}_{{m}} =\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{{m}} =\mathrm{0}\:\:\:\left({m}=\mathrm{1},\mathrm{2},\mathrm{3},…,\mathrm{12}\right) \\ $$$${x}_{\mathrm{13}} =\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{13}} =\mathrm{2} \\ $$$$ \\ $$$${x}_{\mathrm{143}} =\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{143}} \\ $$$${q}_{{r}} ^{\mathrm{143}} =\left({q}_{{r}} ^{\mathrm{13}} \right)^{\mathrm{11}} =\left({q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\right)^{\mathrm{11}} =\underset{{k}=\mathrm{0}} {\overset{\mathrm{11}} {\sum}}{C}_{{k}} ^{\mathrm{11}} {q}_{{r}} ^{\mathrm{4}{k}} \mathrm{2}^{\mathrm{11}−{k}} \\ $$$$ \\ $$$${x}_{\mathrm{143}} =\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} \left(\underset{{k}=\mathrm{0}} {\overset{\mathrm{11}} {\sum}}{C}_{{k}} ^{\mathrm{11}} {q}_{{r}} ^{\mathrm{4}{k}} \mathrm{2}^{\mathrm{11}−{k}} \right) \\ $$$${x}_{\mathrm{143}} =\underset{{k}=\mathrm{0}} {\overset{\mathrm{11}} {\sum}}{C}_{{k}} ^{\mathrm{11}} \mathrm{2}^{\mathrm{11}−{k}} \left(\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{4}{k}} \right) \\ $$$${k}=\mathrm{0}\:\Rightarrow\:\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{4}{k}} =\mathrm{1} \\ $$$${k}=\mathrm{1}..\mathrm{3}\:\Rightarrow\:\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{4}{k}} =\mathrm{0} \\ $$$$\Rightarrow{x}_{\mathrm{134}} =\mathrm{2}^{\mathrm{11}} +\underset{{k}=\mathrm{4}} {\overset{\mathrm{11}} {\sum}}{C}_{{k}} ^{\mathrm{11}} \mathrm{2}^{\mathrm{11}−{k}} \left(\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{4}{k}} \right) \\ $$$${k}=\mathrm{4}..\mathrm{11}: \\ $$$${q}_{{r}} ^{\mathrm{16}} ={q}_{{r}} ^{\mathrm{13}} {q}_{{r}} ^{\mathrm{3}} =\left({q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\right){q}_{{r}} ^{\mathrm{3}} ={q}_{{r}} ^{\mathrm{7}} +\mathrm{2}{q}_{{r}} ^{\mathrm{3}} \\ $$$$\Rightarrow\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{16}} =\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{7}} +\mathrm{2}\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{3}} =\mathrm{0} \\ $$$${q}_{{r}} ^{\mathrm{20}} ={q}_{{r}} ^{\mathrm{13}} {q}_{{r}} ^{\mathrm{7}} =\left({q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\right){q}_{{r}} ^{\mathrm{7}} ={q}_{{r}} ^{\mathrm{11}} +\mathrm{2}{q}_{{r}} ^{\mathrm{7}} \\ $$$$\Rightarrow\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{20}} =\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{11}} +\mathrm{2}\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{7}} =\mathrm{0} \\ $$$${q}_{{r}} ^{\mathrm{24}} ={q}_{{r}} ^{\mathrm{13}} {q}_{{r}} ^{\mathrm{11}} =\left({q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\right){q}_{{r}} ^{\mathrm{11}} =\left({q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\right){q}_{{r}} ^{\mathrm{2}} +\mathrm{2}{q}_{{r}} ^{\mathrm{11}} \\ $$$$\Rightarrow\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{24}} =\mathrm{0} \\ $$$${q}_{{r}} ^{\mathrm{28}} =\left({q}_{{r}} ^{\mathrm{13}} \right)^{\mathrm{2}} {q}_{{r}} ^{\mathrm{2}} =\left({q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\right)^{\mathrm{2}} {q}_{{r}} ^{\mathrm{2}} ={q}_{{r}} ^{\mathrm{10}} +\mathrm{4}{q}_{{r}} ^{\mathrm{6}} +\mathrm{4}{q}_{{r}} ^{\mathrm{2}} \\ $$$$\Rightarrow\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{28}} =\mathrm{0} \\ $$$${q}_{{r}} ^{\mathrm{32}} =\left({q}_{{r}} ^{\mathrm{13}} \right)^{\mathrm{2}} {q}_{{r}} ^{\mathrm{6}} =\left({q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\right)^{\mathrm{2}} {q}_{{r}} ^{\mathrm{6}} ={q}_{{r}} ^{\mathrm{14}} +\mathrm{4}{q}_{{r}} ^{\mathrm{10}} +\mathrm{4}{q}_{{r}} ^{\mathrm{6}} \\ $$$$=\left({q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\right){q}_{{r}} +\mathrm{4}{q}_{{r}} ^{\mathrm{10}} +\mathrm{4}{q}_{{r}} ^{\mathrm{6}} \\ $$$$\Rightarrow\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{32}} =\mathrm{0} \\ $$$${q}_{{r}} ^{\mathrm{36}} =\left({q}_{{r}} ^{\mathrm{13}} \right)^{\mathrm{2}} {q}_{{r}} ^{\mathrm{10}} =\left({q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\right)^{\mathrm{2}} {q}_{{r}} ^{\mathrm{10}} =\left({q}_{{r}} ^{\mathrm{8}} +\mathrm{4}{q}_{{r}} ^{\mathrm{4}} +\mathrm{4}\right){q}_{{r}} ^{\mathrm{10}} \\ $$$$={q}_{{r}} ^{\mathrm{18}} +\mathrm{4}{q}_{{r}} ^{\mathrm{14}} +\mathrm{4}{q}_{{r}} ^{\mathrm{10}} \\ $$$$=\left({q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\right){q}_{{r}} ^{\mathrm{5}} +\mathrm{4}\left({q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\right){q}_{{r}} +\mathrm{4}{q}_{{r}} ^{\mathrm{10}} \\ $$$$\Rightarrow\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{36}} =\mathrm{0} \\ $$$${q}_{{r}} ^{\mathrm{40}} =\left({q}_{{r}} ^{\mathrm{13}} \right)^{\mathrm{3}} {q}_{{r}} =\left({q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\right)^{\mathrm{3}} {q}_{{r}} =\left({q}_{{r}} ^{\mathrm{12}} +\mathrm{6}{q}_{{r}} ^{\mathrm{8}} +\mathrm{12}{q}_{{r}} ^{\mathrm{4}} +\mathrm{8}\right){q}_{{r}} \\ $$$$={q}_{{r}} ^{\mathrm{13}} +\mathrm{6}{q}_{{r}} ^{\mathrm{9}} +\mathrm{12}{q}_{{r}} ^{\mathrm{5}} +\mathrm{8}{q}_{{r}} \: \\ $$$$\Rightarrow\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{40}} =\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{13}} =\mathrm{2} \\ $$$${q}_{{r}} ^{\mathrm{44}} =\left({q}_{{r}} ^{\mathrm{13}} \right)^{\mathrm{3}} {q}_{{r}} ^{\mathrm{5}} =\left({q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\right)^{\mathrm{3}} {q}_{{r}} ^{\mathrm{5}} =\left({q}_{{r}} ^{\mathrm{12}} +\mathrm{6}{q}_{{r}} ^{\mathrm{8}} +\mathrm{12}{q}_{{r}} ^{\mathrm{4}} +\mathrm{8}\right){q}_{{r}} ^{\mathrm{5}} \: \\ $$$$={q}_{{r}} ^{\mathrm{17}} +\mathrm{6}{q}_{{r}} ^{\mathrm{13}} +\mathrm{12}{q}_{{r}} ^{\mathrm{9}} +\mathrm{8}{q}_{{r}} ^{\mathrm{5}} \\ $$$$=\left({q}_{{r}} ^{\mathrm{4}} +\mathrm{2}\right){q}_{{r}} ^{\mathrm{4}} +\mathrm{6}{q}_{{r}} ^{\mathrm{13}} +\mathrm{12}{q}_{{r}} ^{\mathrm{9}} +\mathrm{8}{q}_{{r}} ^{\mathrm{5}} \\ $$$$\Rightarrow\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{44}} =\mathrm{6}\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{13}} =\mathrm{6}×\mathrm{2}=\mathrm{12} \\ $$$${x}_{\mathrm{143}} =\mathrm{2}^{\mathrm{11}} +{C}_{\mathrm{10}} ^{\mathrm{11}} \mathrm{2}^{\mathrm{1}} \left(\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{40}} \right)+{C}_{\mathrm{11}} ^{\mathrm{11}} \mathrm{2}^{\mathrm{0}} \left(\underset{{r}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}{A}_{{r}} {q}_{{r}} ^{\mathrm{44}} \right) \\ $$$${x}_{\mathrm{143}} =\mathrm{2}^{\mathrm{11}} +\mathrm{11}×\mathrm{2}×\mathrm{2}+\mathrm{1}×\mathrm{2}^{\mathrm{0}} ×\mathrm{12} \\ $$$$\Rightarrow{x}_{\mathrm{143}} =\mathrm{2}^{\mathrm{11}} +\mathrm{44}+\mathrm{12}=\mathrm{2104} \\ $$$$ \\ $$$${in}\:{this}\:{way}\:{we}\:{can}\:{calculate}\:{every} \\ $$$${term}\:{of}\:{the}\:{sequence}\:{even}\:{though} \\ $$$${we}\:{don}'{t}\:{know}\:{the}\:{values}\:{of}\:{A}_{{r}} \:{and} \\ $$$${q}_{{r}} \:\left({r}=\mathrm{1},\mathrm{2},…,\mathrm{13}\right). \\ $$

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