(Jên+19+S6nk )II 76ip+S6i-1h Therefore п-1 foi+aP + fei+3h' I6n-3 = k11( fei+3P + fei+2h ) föiq + fei-1k fei-19+ fei-2k i=0 Also, from Eq.(8), we see that X6n-4X6n-5 X6n-2 = X6n-5 + X6n-4 + x6n-7 n-2...

Show me the steps of determine purple and information is here step by step

Extracted text: (Jên+19+S6nk )II 76ip+S6i-1h Therefore п-1 foi+aP + fei+3h' I6n-3 = k11( fei+3P + fei+2h ) föiq + fei-1k fei-19+ fei-2k i=0 Also, from Eq.(8), we see that X6n-4X6n-5 X6n-2 = X6n-5 + X6n-4 + x6n-7 n-2 fei+8p+ f6i+7h' fei+7P + fei+ch fei+49+ fei+3k fei+39 + fei+2k, 2р + h p+h i=0 n-1 n-2 ( 2pth) pth f6i+8P+S6i+7h)((si+49+/6i+3k i=0 i=0 + n-2 n- I e 6i+29+16i+1k)+pT(6it+6P+S6i+5h\(f6i+29+S6i+1* Tôi+19+S6ik ) S6iP+S6i-1h T6i+19+S6ik Ssi+5P+S6i+4h , i=0 i=0 foi+8P + fei+7h Jõi+7p + feineh )( 6i+49 + foi+3k п-2 2р + h = r p+h foi+39 + f6i+2k / i=0 n-2 2p+h S6i+8P+SBi+7h S6i+49+S6i+3k i=0 n-1 n-1 n-2 f6n9+Sên-1k ( S6i-1P+f6i-2hY fên+19+f6nk) f6ip+f6i-1h +p ( f6i+6P+S6i+5hY i=0 i=0 i=0 п-2 (2p + h' fei+8P + f6i+7h fei+49 + fei+3k П fei+7P+ fei+sh) foi+39 + fei+2k, p+h i=0 n-2 h r 2p+h t 11+7p+ fei+sh) (Tei+39+Ssi+2k fBi+8p+ fei+7h fei+49+S6i+3k p+h i=0 n-1 n-2 | f6i-1p+fsi-2h I foi+6p+fei+5h fên9+Sen-1k Jên+19+Sönk / i=0 n-2 h+P i=0 n-1 | fsi+5p+ f6i+4h foip+Sei-1h i=0 i=0 15 n-2 fei+49 + fei+3k\ foi+39 + f6i+2k / 2р + h (foi+sP + f6i+7h` II X6n-2 p+h foi+7P + foi+sh, i=0 n-2 A() II ( ( 2p+h P+h foit49+S6i+3k Sei+39+ Sei+2k Fes+7p+fei+eh) i=0 feng+fen-ik h h +p(Ton+1q+fönk n-2 foi+sP+ foi+7h II foi+7p+ fei+6h, fei+49 + foi+3k\ f6i+39 + fei+2k, 2р + h X6n-2 p+h i=0 hr() II ( ) (f n-2 f6i+8p+fei+7h fei+7p+f6i+6h i=0 f6i+49+f6i+3k Jei+39+S6i+2k hrpth fönq+ fen-1k fên +19+ fenk h n-2 fei+49 + fei+3k fei+39 + f6i+2k, 2р + h (foi+sp+ fei+7h II foi+7P + foi+6h, X6n-2 p+h i=0 п-2 2p+h p+h )II f6i+sp+ fei+7h Jei+7p+f6i+sh) i=0 f6i+49+S6i+3k Sei+39+f6i+2k +- fên+19+fénk+fenq+fôn-1k fen+19+fenk n-2 fei+8P + fei+7h( foi+49 + fei+3k foi+7P + fei+6h, 2р + П = r p+h foi+39 + fei+2k, i=0 п-2 2p+h T(fsi+8P+f6i+7h p+h fei+49+S6i+3k fei+39+ fei+2k foi+7p+fei+sh) i=0 fön+29+fen+1k fon+19+fenk n-2 2р + h II fei+8P + fei+7h` foi+7P + fei+6h, fei+a9+ foi+3k foi+39 + fei+2k fon+19 + fönk fen+29 + fön+1k p+h i=0 п-2 fei+8P+ f6i+7h foi+7P + fei+6h, foi+49+ fei+3k f6i+39 + fei+2k, fön+29 + fon+1k + fen+19 + fenk] fon+29 + fon+1k 2р + h = r p+h i=0 п-2 fei+8P + f6i+7h` П fei+7p + fei+6h) (foi+39 + fei+2k ) [fön+2g + fon+1k 2p +h foi+49+ foi+3k\ [fon+39 + fon+2k] = r p+h i=0 16


Extracted text: Brn-1In-2 YIn-1 + &xn-4 In+1 = a1n-2 + n = 0, 1,..., (1) 1 The following special case of Eq.(1) has been studied In-1In-2 Int1 = In-2+ (8) Tn-1 + In-4 where the initial conditions r-4, I-3, T-2, T-1,and xo are arbitrary non zero real numbers. Theorem 4. Let {zn}-4 be a solution of Eq.-(8). Then for n = 0, 1,2, ... fep + fes-ih ( foi+29 + foiik Jei-1p+ fei-zh) fo419 + Sauk) farq + Sai-1k feirap + forah ) To-19 + foi-ak , (Sei+aP + fot+3h) fei-2P + fei+1h (foi+19 + Sei+ak = rT fei+1P+ fesh ) Joi+39 + fei+2k ) (fei+ep + fes+sh\ (foi+29 + fei+ik Soi-sp + feisah) fei+ap + fei+ah\ ( feiseq + fei-sk Sai+ap + fei+2h ) Jai459 + Seisak ) fei+19 + fask 2p +h II ap+ fesch) (Tars9 + Suvzk, foi+sP + fouth ( Seisa9 + Soirak In+1 = p+h where r-4 = h, r-3 = k, z-2 = r, r-1 = P, 2o = q, {fm}-1 = {1,0,1,1,2, 3, 5, 8, .}. Proof: For n -0 the result holds. Now suppose that n>0 and that our assumption holds for n - 2. That is; feirap + foi+ah Jei+ap + fei42h) J6i-19 + fei-2k fauq + foi-1 k i-0 fai+1p + fesh ) fet439 + feirak) Joi+ep + fei+sh (fei+29 + foi+zk foi+sp + fes+ah) Iom-6 = 9TT(fei-4p + foi+3h (fei+09 + fei-sk + fei+zh) \Torr39 + fei+ak ) * (2p + h ( foi-sp+ fesuzh (foir49 + foi+ak Ji+7p + Soisgh) Tausg + Soi+zk p+h Now, it follows from Eq.(8) that IGn-4 = Iộn-7 + 11 feisep + fo+sh ( foi+29 + fe+ik Sunsp + Seissh ) (Tus1g+ fesk) = p|| •II )( )II( ) ) (foi+op + fou-sh ( fei-29 + fei+ikY 1foi+sp + foi+sh) foi+19 + fosk + A PIT fasop + foish) ( for-29 + Sousik Joi4sp+fostah) n-2 = p Fei+sp + feirah) feir19 + faik pg ( foi+6P + fei+sh) ( fei+29 + fei-ik =PlI(asp+ fash)( Far1q + Sak ) 12 PII ´ fouvap + faunh\ ( fa-29 + feieik) Sos+19 + Souk PIIusp + Jourah) 1+ ( PII( ) ) -PT (furep + fouosh (fau29 + fuosk Sei-sp + foi-gh ) fois19 + fuk) n-2 PII( (unt Jeusp + Sanah) Sesig+ fauk (An ( fesep + fursh ( fev29 + Jairak1+a+ fon-ek - PII Sen-o9 + fon-zk] n-2 Sen-s9 + Son -ak-" Sei+ep + Sei+sh) ( Su-29 + Sai+ik [ Sen-sq + Sen-sk = PII(sp+ fursh)Tuer9+ feck ) [Ton-39 + Sen-sk] Therefore R-1 Also, from Eq.(8), we see that Tusap + Seurzli ) ( Jonnog + Senek ) ( )--II( )( ) 13