ELEMENTARY PROBLEMS AND SOLUTIONS

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1 ELEMENTARY PROBLEMS AND SOLUTIONS EDITED BY So L. BASIN, SYLVANIA ELECTRONIC SYSTEMS, MT. VIEW, CALIF. Sed all commucatos regardg Elemetary Problems Solutos to S e L 8 Bas, 946 Rose Ave., Redwood Cty, Calfora. We welcome ay problems beleved to be ew the area of recurret sequeces as well as ew approaches to exstg problems. The proposer must submt hs problem wth soluto legble form, preferably typed double spacg, wth ame(s) address of the proposer clearly dcated. Solutos should be submtted wth two moths of the appearace of the problems. B-24 Proposed by Brother U.Alfred,St.Mary's College, Calf, It s evdet that the determat I F F F \ T? Tp T? J F F F I has a value of zero. Prove that f the same quatty k s added to each elemet of the above determat, the value becomes (-1) k. s 95 Proposed by Brother U, Alfred. Fd a expresso for the geeral term(s) of the sequece T = 1, Tj = a, T 2 = a, where T 2 = YTZ "* 2-2 T = T 2 T 2-l B-26 Proposed by 5.L-Bas,SyIvaa Electroc Systems,Mt.Vew,Calf. Gve polyomals b (x) B (x) defed by b (x) = 1, B (x) = 1 b (x) = xb f l - 1 (x) + b _ 1 (x) ( > 1) B (x) = (x + 1) B + 1 (x) + b _ 1 (x) ( > 1) 73

2 74 ELEMENTARY PROBLEMS AND SOLUTIONS [Dec. show that b (x) = P (x) v ' 2 ' where M B (x) = P ^ (x) ' 2 + l v ' >.«- [ I ] (V). [^ beg the greatest teger l e s s tha or equal to -r-. B-27 Proposed by D.G. Cross, Brmgham, Egl* Let x = cos, [ z ] s the greatest teger cotaed z. cos = x cos 2 = 2x 2-1 cos 3 = 4X 3-3x cos 4 = 8x 4 ~ 8x cos 5 = 16x 5-2x 3 + 5x cos 6 = 32x 6-48x x 2-1 N cos = P (x) = 2 A. x + 2 ~ 2 j (N = [ ( + l ) / 2 ] s J 3 = 1 g r e a t e s t teger fucto.) Show () V = 2 ( = > V,+1 2 V,- A j,( = N - 1 ) () P + 2 ( x ) = 2 x P + 1 ( x ) - P (x) N (v) v If A = 2 A. I, the A ^ = 2A _ + A. '. -.1 l» =1 J Note: (A t = 1, A 2 = 3, 7 = A 3 = 2A 2 + A t = ). B-28 Proposed by Brother U. Alfred, Usg the e Fboacc umbers F 2 to F 1 ( 1, 2, 3, 5, 8, 1 3, 2 1, 3 4, 5 5 ),

3 1963] ELEMENTARY PROBLEMS AND SOLUTIONS 75 determe a thrd-order determat havg each of these umbers as elemets so that the value of the determat s a maxmum. B Proposed by A.P. Boblett, U.S. Naval Ordace Laboratory, Coroa, Calfora. Defe a geeral Fboacc sequece such that F t 1 = a; F 2 L = b; s ' F = F -2 + F -1 - > Also defe a characterstc umber, C, for t h s sequece, where C = (a + b)(a - b) + ab. Prove: F,, F, - F 2 = (-l) C, for all ' SOLUTIONS Solutos to Problems B6 B9 through B15, Vol. 1, No. 2, Aprl, 1963 SOME REFLECTIONS B - 6 Proposed by Leo Moser, Uversty of Alberta, Edmoto, Alberta, Lght rays fall upo a stack of two parallel plates of glass, oe ray goes through wthout reflecto, two rays (oe from each terval terface opposg the ray) wll be reflected oce but dfferet ways, three wll be reflected twce but dfferet ways. Show that the umber of dstct paths, whch are reflected exactly tmes, s F _. Soluto by J. L. Brow, Jr., Pesylvaa State Uversty, Pesylvaa All rays whch experece exactly reflectos wll emerge from the same face, ether top or bottom of the stack; furthermore, f those havg - 1 reflectos emerge from the top face, the those havg reflectos wll emerge from the bottom face. Let us assume, wthout loss of geeralty that the rays havg exactly reflectos wll emerge from the bottom face as show below for the case of two reflectos.

4 76 ELEMENTARY PROBLEMS AND SOLUTIONS [Dec. Let a be the umber of dstct paths whch have exactly reflectos. If we cosder ay emerget ray whch has had reflectos ( ^ 2), the t m u s t have had ts last, or reflecto from ether face or terface 1. The umber of dstct paths havg the reflecto at face s equal to the umber of dstct paths reachg face after - 1 reflectos, or a -. Smlarly, the paths whose reflecto s at terface 1 must have had the ( - l)th reflecto at face 2, the umber of dstct paths s the equal to the umber of dstct paths reachg face 2 after ( - 2) reflectos, or a o. Sce the two possbltes a r e mutually exclusve exhaustve, we ^ have a for ^ 2. The tal codtos, a Q = 1, a t = 2 establsh that a = F for. DrZ FIBONACCI SUMS B-9 Proposed by R.L. Graham, Bell Telephoe Laboratores.Murray Hll,N.J. Prove - F F = 1 = th where F s the = Fboacc umber. B-9 Soluto by F r a c s D. P a r k e r, Uversty of Alaska. the I Sce ^ -^ -^ +1-1 F F F F F F F F l = F, F - 1 F F t 1 +1 JL _ JL L l 1-2 F, F - 1 J 1^ F F _ +1 Smlarly, F F TP F = "-1 F F " l 1" sj. 2 + "I r 1 3_ + 2-3~3.5 " "1 + "1 1* 2 5_^ 3 8_ = 2

5 1963] ELEMENTARY PROBLEMS AND SOLUTIONS 77 Edtoral Commet: The above soluto to problem B- 9 s a good example of a prcple foud may other problems umber theory, amely formg a sum, t s ofte helpful to judcously group the terms a certa fasho. A example of ths may be foud provg the followg theorem cocerg the dvsor fucto r(). Prove r(r) s odd f oly f s a square. LUCAS-FIBONACCI IDENTITY B-1 Proposed by Stephe Fsk f 'Sa Fracsco, Calfora.. Prove the de Movre-type" detty, L + V5F \ P L + \/5F \ _ p p 2 ) 2 where L deotes the th Lucas umber F deotes the th Fboacc umber. B-1 Soluto by Charles Wall, Ft. Worth, Texas. Sce T f r T, L + v5 F,, _ a + (3 + a - [3 where 2 ~ 2 + 'sts R. - \Ts 2 ' P 2 we have L + s/5 F \ _ P _ p p, ^p, p a^ + p^ + a^-f^ p L + 'Vs F p p_ ~ a 2 2 B - l l Proposed by S,L.Bas,Sylvaa 'Electroc Defese Laboratory, Show that the hypergeometrc fucto rt \ - V 2 k ( + k)? (x - l ) k ~ k= fa-k- 1 )-' (2k+ 1)! geerates the sequece G ( f ) = F 2' = 1-2 ' 5 ''-- B - l l Soluto by S. L. Bas, Sylvaa Electroc Systems, Mouta Vew, Calfora Sa Jose State College v 2 ( + k)! (x - 1) ( k k= ' " 1 )! ( 2 k + 1 )! " where U - (x) are the Chebyshev polyomals of the secod kd _ 1

6 78 ELEMENTARY PROBLEMS AND SOLUTIONS [Dec. U,(x) - r W^T7 {(x + <Jx2 - f - (x - ^ x 2 - l ) } "-(DvU 3 + 4~5\ U Observg that we have _ (3 - \T5 -s). (^)! (^a). (. 41 ^(D-M^f Commet: Settg x = 3/2, ( + k)! 2 r 2 k = Q (2k + 1)! ( - k - 1). ~ k = Q )') the summato becomes \T 2 j ( 2 + k ( 2k + 1 )~ F 2 ' Rsg dagoals\ of P a s c a l ' s j tragle / See Fg. 1, page 24, October, 1963, Fboacc Quarterly. A LUCAS DETERMINANT B-12 Proposed by Paul F. Byrd, Sa Jose State College, Sa Jose, Calf. Show that 3 V ' V 1 ^ where L s the th Lucas umber gve by L< = 1, L 2 = 3, L l = L t 1 to J l L L, = N/~-1. ' B-12 Soluto by Marjore Bckell, Sa J o s e State College, Sa J o s e, Calf. Let D deote the determat of o r d e r. Expg the determat by ts th row we have. D = D - + D wth D 1 = 3, Do = 4 so that D J x L ' * Also solved by Wllam A. Beyer, Los Alamos, New Mexco

7 1963 J ELEMENTARY PROBLEMS AND SOLUTIONS 7 9 B-13 Proposed by S.L, Bas, FIBONACCI CONTINUANT Determats of o r d e r whch a r e of the form, K ( b, e, a ) - c a b c a b c a b c a a r e kow a s CONTINUANTS Prove that K ( b, c, a ) = (c + ^ c 2-7 a b ) (c - * a b ) Vc 2-4 ab show, for specal values of a, b, c, that K ( b, c, a ) = F B-13 Soluto by Marjore Bckell, Sa Jose State College, Sa J o s e, Calf. Expg K ( b, c, a ) by the th row we obta, (1) K ( b, c, a ) = c K _ 1 ( b, c, a ) - a b K _ 2 ( b, c, a) If u v a r e the roots of the quadratc equato x 2 - ex + ab =, the (2) u = Q"(C' + ^ c 2-4 a b ), v = (c - Vc 2-4ab) Now K ( b, c, a ) = (u - v ) / (u - v) by ducto K (b,c,a) = F - for values of a, b, c whch yeld the quadratc x 2 - x - 1,. e., a = c = 1, b = - 1 ; a = - 1 b = c = 1; a = b = = N/-1 c = 1. A LITTLE SURPRISE B_14 Proposed by Maxey Brooke, Sweey, Texas C.R. Wall,Ft.Worth,Tex Show that OO - p ( - l ) + 1 F. V 1 -. y 1 2, = - ^ 19 =l 1 =l 1 B-14 Soluto by Charles Wall, Ft. Worth, Texas Sce 2 F x 11 = =l 1 - x - x2

8 8 ELEMENTARY PROBLEMS AND SOLUTIONS [Dec. 1963] the =l r. ) L ^ -* ) J =l Also solved by Dermott A. Breault, Sylvaa, ARL, Waltham, Mass. FIBONACCI SEQUENCE PERIODS B Proposed by R.B.Wallace, Beverly Hll s, Call f'.. Stephe Geller, Uversty of Alaska, College, Alaska. If p, s the s m a l l e s t postve teger sueh that F ^ = F mod (lo 1 *); +p k for all postve, the p, s called the perod of the Fboacc sequece k relatve to 1. Show that p, exsts for each k, fd a specfc formula for p as a fucto of k. Edtoral Commet: Ths problem s dscussed ths ssue a paper by Dov J a r d e whch s a reply to Stephe Geller f s letter to the edtor, p e 84, Aprl, 1963, Fboacc Quarterly, EDITORIAL ASSOCIATES (Cot.) well as those who have the teto of dog so 9 wll receve recogto as Edtoral Assocates. The Edtor should be cotacted by ayoe who wshes to be assocated wth the Fboacc Q u a r - terly ths maer. R E N E W Y O U R S U B S C R I P T I O N!!!