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sol_ch8_recurrence

# sol_ch8_recurrence - .kdswhu Uhfxuuhqfh Uhodwlrqv;14 41...

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Fkdswhu ; Uhfxuuhqfh Uhodwlrqv ;14 Vhtxhqfhv dqg Uhfxuuhqfh Uhodwlrqv 41 +d, d 3 @4 >d 4 @ d 3 .7 @ 4.7 @ 8 5 @ d 4 .7 @ 8.7 @ < 6 @ d 5 .7 @ <.7 @ 46 > d 7 @ d 6 = +e, d 3 @8 4 @ d 3 .5 5 @ d 4 6 @ d 5 > d 7 @ d 6 [email protected]:.;@58 = +f, d 3 4 @5 d 3 .4 5 4.4 @ 6 5 d 4 5 [email protected] 6 d 5 .6 5 43.< 5 @5< > d 7 d 6 .7 5 = +g, d 3 4 5 @ d 4 d 3 @5.7 [email protected] 6 @ d 5 d 4 @9.7 [email protected] 7 @ d 6 d 5 @ 47.7 = +h, d 3 4 > d 5 @: d 4 45 d 3 5 45 [email protected] 73 > d 6 d 5 45 d 4 + 73, 45 [email protected] 5<; > d 7 d 6 45 d 5 + 5<;, 45 + 4933 = +i, d 3 @ 4 4 > d 5 @ d 4 . d 3 [email protected] [email protected] > d 6 @ d 5 . d 4 > d 7 @ d 6 . d 5 = +j, d 3 4 5 > d 6 d 5 . d 4 . d 3 [email protected] > d 7 d 6 . d 5 . d 4 [email protected] = 61 d 3 4 5 @3 6 @ d 5 . d 3 7 @ d 6 . d 4 8 @ d 7 . d 5 > d 9 @ d 8 . d 6 : @ d 9 . d 7 = 444

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445 ;1 Uhfxuuhqfh Uhodwlrqv 81 d 3 @3 >d 4 @4 >=== > d q 4 @5 q 4 4 q q 4 >=== = Qrz d q q [email protected] 5 q 4 5 q 4 q 4 d q 4 .4 = Khqfh/ wkh uhfxuuhqfh uhodwlrq lv d q @ d q 4 >q 5 > zlwk wkh lqlwldo frqglwlrq d 4 = :1 d 4 > === > d q 4 @ q 4 q @ q> d q .4 @ q Wkxv/ d q @ q @ q [email protected] d q 4 = Khqfh/ wkh uhfxuuhqfh uhodwlrq lv d q @ d q 4 5 > zlwk wkh lqlwldo frqglwlrq d 4 = <1 Ohw d q ghqrwh wkh xqsdlg edodqfh diwhu q sd|phqwv1 Qrwh wkdw wkh xqsdlg edodqfh diwhu q sd|phqwv lv wkh xqsdlg edodqfh diwhu q 4 sd|phqwv soxv wkh prqwko| lqwhuhvw plqxv rqh sd|phqw1 Wkxv/ d q @ d q 4 . 3 = 3< 433 d q 4 7333 = Wkdw lv/ d q @ d q 4 .3 = 333< d q 4 7333 = 333< d q 4 7333 4 = Wkh lqlwldo xqsdlg edodqfh lv wkh frvw ri wkh krxvh plqxv wkh grzq sd|phqw/ l1h1/ '773 > 333 = Wkxv/ wkh lqlwldo frqglwlrq lv d 3 @ 773333 = 441 +d, Wkh dprxqw wkdw dffxpxodwhv diwhu rqh |hdu lv wkh lqlwldo dprxqw soxv wkh lqwhuhvw rq wkh lqlwldo dprxqw1 Qrz D q 4 lv wkh dprxqw dffxpxodwhg diwhu q 4 |hduv1 Wklv lpsolhv wkdw wkh dprxqw dw wkh ehjlqqlqj ri q wk |hdu lv D q 4 = Lw iroorzv wkdw wkh wrwdo dprxqw dffxpxodwhg diwhu q |hduv lv wkh dprxqw dw wkh ehjlqqlqj ri wkh q wk |hdu soxv wkh lqwhuhvw rq wkdw dprxqw1 Ehfdxvh wkh lqwhuhvw udwh lv : = 8( > wkh lqwhuhvw hduqhg gxulqj wkh q wk |hdu lv +3 = 3:8, D q 4 = Khqfh/ D q @ D q 4 . +3 = 3:8, D q 4 = 3:8 D q 4 4 = Dovr D 3 @:333 = Wkxv/ zh qg wkdw d uhfxuuhqfh uhodwlrq dqg wkh lqlwldo frqglwlrq iru wkh vhtxhqfh i D q j " q @3 duh D q = 3:8 D q 4 4 > D 3 = +e, D 3 > D 4 = 3:8 D 3 = 3:8 :333 @ :858 > D 5 = 3:8 D 4 = 3:8 :858 @ ;3;< = 6: > D 6 = 3:8 D 5 = 3:8 ;3;< = 6: @ ;9<9 = 3: = D 7 = 3:8 D 6 = 3:8 ;9<9 = 3: @ <67; = 5: = D 8 = 3:8 D 6 = 3:8 <67; = 5: @ 4337< = 6< = Wkh wrwdo dprxqw dffxpxodwhg diwhu 8 |hduv lv '4337< = 6< =
;141 Vhtxhqfhv dqg Uhfxuuhqfh Uhodwlrqv 446 461 Ohw E q @ wkh qxpehu ri q 0elw elqdu| vwulqjv wkdw gr qrw frqwdlq 44 dv d vxevwulqj1 Qrz 3 dqg 4 duh wkh rqo| 4 0elw elqdu| vwulqjv dqg wkhvh vwulqjv gr qrw frqwdlq 44 > vr E 4 @5 1 Qh{w/ 33 > 34 > 43 > 44 duh wkh rqo| 5 0elw elqdu| vwulqjv1 Krzhyhu/ dprqj wkhvh 34 > 43 > 33 duh wkh rqo| 5 0elw elqdu| vwulqjv wkdw gr qrw frqwdlq 44 = Wkhuhiruh/ E 5 @6 = Qh{w zh qg wkdw 333 > 334 > 343 > 433 > 434 > 344 > 443 > 444 duh wkh rqo| 6 0elw vwulqjv1 Dprqj wkhvh 333 > 334 > 343 > 433 > dqg 434 duh wkh rqo| 6 0elw elqdu| vwulqjv wkdw gr qrw frqwdlq 44 = Wkxv/ E 6 @8 = Khqfh/ E 4 >E 5 > dqg E 6 = Ohw xv qrz qg d uhfxuuhqfh uhodwlrq dqg lqlwldo frqglwlrqv iru wkh vhtxhqfh E 4 5 6 7 >=== = Zh dvvxph q 6 = Wr frxqw wkh qxpehu ri q 0elw elqdu| vwulqjv wkdw gr qrw frqwdlq 44 dv d vxevwulqj zh frqvlghu wkh iroorzlqj wzr w|shv ri vwulqjv= +l, q 0elw elqdu| vwulqjv wkdw ehjlq zlwk 3 dqg wkdw gr qrw frqwdlq 44 dv d vxevwulqj/ +ll, q 0elw elqdu| vwulqjv wkdw ehjlq zlwk 4 dqg wkdw gr qrw frqwdlq 44 dv d vxevwulqj1

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