LINEAR RECURSION RELATIONS - LESSON FOUR SECOND-ORDER LINEAR RECURSION RELATIONS

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1 LINEAR RECURSION RELATIONS - LESSON FOUR SECOND-ORDER LINEAR RECURSION RELATIONS BROTHER ALFRED BROUSSEAU St. Mary's College, Califoria Give a secod-order liear recursio relatio (.1) T. 1 = a T + b T 1, where a ad b are real umbers ad the values T. of the sequece are real as well, there is a auxiliary equatio: (2) x 2 - ax - b = 0, with roots (3)' a + Va 2 + 4b" 2 a - Va b As is usual with quadratic equatios, three cases may arise depedig o whether a 2 + 4b > 0, roots real ad distict; (4) a 2 + 4b = 0 S roots real ad equal; a 2 + 4b < 0, roots complex umbers. CASE 1. a 2 4-4b > 0. I previous lessos we have cosidered cases of this kid. It has bee oted that the roots may be ratioal or irratioal. There seems to be othig to add for the momet to the discussio of these cases. 194

2 Apr SECOND-ORDER LINEAR RECURSION RELATIONS 195 CASE 2. a 2 + 4b, = 0. The presece of multiple roots i the auxiliary equatio clearly requires some modificatio i the previous developmet. If x 2 - ax - fo = 0-1, -2 A x - ax - bx = 0. Sice the equatio has a multiple root (a/2), will have this same root. Hece the derivative of this equatio (5) x 11 " 1 - a( - l)x " 2 - b( - 2)x ~ 3 = 0 is satisfied by the multiple root also. Thus the multiple root, r, satisfies the followig two relatios: (6) -1,, -2 r = ar + br t -v - 1,, t «v -2 r = a( - l)r +b(-2)r The result is that if we formulate T as (7) T = A r + B r m A /, i\ +1, -o +1 T - = A(+l)r +Br it follows that (8) T. = a T, 1 + b T = A [a( + l)r + b r A t, 0v +2, _, +2 = A( + 2)r +B r +1,, i

3 196 SECOND-ORDER LINEAR RECURSION RELATIONS [Apr. so that the form of T is iaitaied. EXAMPLE Fid the expressio for T i terms of the roots of the auxiliary equatio correspodig to the liear recursio relatio T _,, = 6T - 9T, if T A = 4, T 2 = 7. double root of 3. Hece T Here the auxiliary equatio is x 2-6x + 9 = 0 with a has the form T = Ax3 + Bx3. Usig the values of Tj ad T 2 4 = Ax3 + Bx3 7 = 2Ax3 2 + Bx3 2 with solutios A = -5/9, B = 17/9. Hece T = :-5x3 + 17x3" = 3-2 1? ) 9 L^ J It may be oted that for ay o-zero multiple root r, coefficiets i the set of equatios for T A ad T^ is the determiat of the r 2 r r = - r 3 which is ot zero, so that these equatios will always have a solutio. CASE 3. a 2 + 4b < 0. The case of complex roots is quite similar to that of real ad distict roots as far as determiig coefficiets from iitial value equatios is co-

4 1969] SECOND-ORDER LINEAR RECURSION RELATIONS 197 cered. However, sice we have specified that the t e r m s of the sequece ad the coefficiets i the recursio relatio a r e r e a l, there will have to be a special relatio betwee A ad B i the expressio for T : T = A r + B s. Note that r ad s a r e complex cojugates, so that r ad s a r e of the form P + Qi ad P - Qi respectively, where P ad Q a r e real. If T is to be r e a l, the A ad B must be complex cojugates a s well. EXAMPLE Fid the expressio for T i t e r m s of the roots of the auxiliary equatio for the liear recursio relatio T 1 = 3 T - 4 T 1, +1-1 with Tj = 5, T 2 = 9. Here the auxiliary equatio i s : x 2-3x + 4 = 0 with roots 3 + i V i *J1 r _, s 2 The 5 = A r + Bs 9 = A r 2 +Bis2 from which oe fids that A = 21 - H i JS/7 ^ = 21 + H i AJ7 A 28 ' 28

5 198 SECOND-ORDER LINEAR RECURSION RELATIONS [Apr. Accordigly, L f21-11w7 \ _, / N/7 \ " \ "28 / r I 28 J 3 AN ANALOGUE Because of the similarities amog secod-order liear recursio relatios it is possible to fid close aalogues amog them to the Fiboacci ad Lucas sequeces. Let us cosider as a example the secod-order liear r e - cursio relatio The auxiliary equatio is T x 1 = 3 T + T'. x2 _ 3 x _ 1 = 0 with roots \/13 _ 1 - N/T r, s -. If the iitial terms are take as T 0 = 0, T 4 = 1, T 2 = 3 S the 1 = Ar + Bs 3 = Ar 2 + Bs 2, with resultig values A = 1/ N/T3 ad B = - 1 / AJ13 SO that T _ r - s _ r - s -s/is r " s has precisely the same form as the expressio for F with 13 replacig 5 uder the square root sig.

6 1969] SECOND-ORDER LINEAR RECURSION RELATIONS 199 If the relatio V^ = T ^ + T _ 1 is used to defie the correspodig,? Lucas" sequece, the terms of this sequece are: V 0 = 2, Vi = 3, V 2 = 11, V 3 = 36,. Solvig for A ad B from 3 = Ar + Bs 11 = Ar 2 + Bs 2 gives values of A = 1, B = 1, so that V = r + s i perfect correspodece to the expressio for the Lucas sequece. As a r e - sult of this similarity, may relatios i the Fiboacci-Lucas complex ca be take over (sometimes with the slight modificatio of replacig 5 by 13) to this pair of sequeces. Thus: T 0 = T V 2 T = T 2 4- T^ x 2+l +1 T,T - - T 2 = {-if"" V 0 = V 2 + 2{-l) V + V _,_ 0 = 13T _,, V 2 + V 2 = 13 (T 2 + T 2 ) PROBLEMS 1. For the sequece T = 1, T 2 = 3, obeyig the liear recursio relatio T-,- = 3 T + T - show that every iteger divides a ifiity of members of the sequece.

7 200 SECOND-ORDER LINEAR RECURSION RELATIONS Apr For the correspodig rt Lucas sequece, prove that if m divides, where is odd, the V divides V. m 3. Fid the expressio for the sequece Tj = 2, T 2 = 5 i terms of the roots of the auxiliary equatio correspodig to the liear recursio relatio r T = 4 T + 4 T.,. 4. Prove that the secod-order liear recursio relatio defies a arithmetic progressio. T ± 1 = 2T - T - 5. If Ti = a, T 2 = b, fid the expressio for T i terms of the roots of the auxiliary equatio correspodig to T - = 4T - 4T If Tj = i, T 2 = 1 ad T - = -T -, fid the geeral expressio for T i terms of the roots of the auxiliary equatio. 7. T t = 3, T 2 = 7, T 3 = 17, T 4 = 43, T 5 = 113,- areterms of a secodorder liear recursio relatio. Fid this relatio ad express T i terms of the roots of the auxiliary equatio* 8. For the secod-order liear recursio relatio T, 1 = 5 T + T - fid the particular sequeces aalogous to the Fiboacci ad Lucas sequeces ad express their terms as fuctios of the roots of the auxiliary equatio. 9. For Ti = 5, T 2 = 9, T ^ = 3T - 5T -, fid T i terms of the roots 1 4 of the auxiliary equatio. 10. If ( A / 3 3 " V 5 + *J~33\ ^ / ^ 3 3 \ / s - /s/33\ \ 33 ) \ 2 J \ 33 ) \ 2 J determie the recursio relatio obeyed by T ad fid Tj ad T 2. [ See page 210 for Solutios to Problems. ]