# SUMS OF n-th POWERS OF ROOTS OF A GIVEN QUADRATIC EQUATION. N.A. Draim, Ventura, Calif., and Marjorie Bicknell Wilcox High School, Santa Clara, Calif.

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3 QUADRATIC EQUATION April TABLE : r * + r r l + r =,, r T r =, r l + r =, -t- r = r l + r =, r l + r = 8, 8 r l + r = 8 + q + pq + p q + q + p q + :pq' + p q + 9p q + q + p q + p q + pq + 8p q + p q + lp q + q., -Z, ( - ) -, ( - )( - ) - r l = p q q p q ~T! T ( - m)( - m - )... ( - m + ) -m+ m ( m - )! q TABLE : C(m, ) m L SC(m, ) L L L L L L 9 8 L 9 L 8

4 9 SUMS OF -th OWERS OF ROOTS OF A GIVEN F r o m the above tables, (A. ). r + l r [a/] i= z^: ) - ( - i - ) s ' -i i p q where [ x ] is the greatest iteger less tha or equal to x, ad ( ) is the biomial coefficiet roof: By algebra, (m-)l! k+ k+ k, k+ A k+ k, k x, k t k. r, + r = (r, + r~ + r r 9 + A r l ^ ' ) - ( ^r, + r, ^ )' = (r^ + r ^ ) ( r i + r ) - r ^ r * " + r k " l ) = p(r + r ) + q(r + r ). 'I F r o m Table, (A. ) holds for =,,..., 8. To prove (A. ) by mathematical iductio, a s s u m e that (A ) holds whe = k ad = k -. Usig the result just give ad the iductive hypothesis, [k/] k+, k+ v r, + r = p i= [(k-l)/] + q i= ( k :S. ( k_i " ) i i k-i i q k - l - i _ k-i- I I k - l - i i q Multiplyig as idicated ad usig the idex substitutio i- for i i the secod s e r i e s yields k+, k+ r + l r [V] i= ( k : i ) - ( k " i _ ) ' k+l-i i q +l)/] + i [( k :j> - ( k ; ; ) i= k+l-i i q

5 QUADRATIC EQUATION April Sice the r e c u r s i o formula for biomial coefficiets is m m m+ l ' l + l ; { + l ' ' combiig the s e r i e s above will yield [ ( k + i - i } _ ( k - i } ] for the coefficiet of p q. Cosiderig the s e r i e s for k eve ad for k odd leads to [(k+l)/] k+, k+ v r x + r = I i= ' ( k + l - i j _ (k-i } ' k+l-i i p q so that (A. ) follows by mathematical iductio. A expressio equivalet to (A. ) was give by Basi i [! ] Usig formula (A. ), we ca write the desired equatio,... v f(x) = x - (r x + r ) x + (-) q =. Example: Give the equatio x - x + =, write the equatio whose roots a r e the fourth powers of the roots of the give equatio, without solvig the give equatio. I the give equatio, p =, q =. F r o m Table or formula (A. ), r^ + rjj = () + () (-) + (-) = 9, so f(x) = x - 9x + =. As a check, by factorig x -x + =, r ] = ad r_ =, givig us f(x) = (x - )(x - ) = x - 9x +. Returig to Table, the summatio of the coefficiets by rows yields the s e r i e s,,,,, 8, 9,,..., the successive Lucas umbers defied by L, =, L =, ad L = L, + L. - - The geeral equatio for r, + r? reduces to the umerical values of C(m, ) of Table for p = q = i the primitive equatio x - px -q = c p, which becomes x - x - = with roots a = ( + /ft)/ ad ( = ( - \/~)/. This yields a expressio for the th Lucas umber, sice L = a + ;

6 9 SUMS OF -th OWERS OF ROOTS OF A GIVEN (A. ) L^ = + + (N-)/! + (-)(-)/! + (-)(-)(-)/! [/] i= ( : i ) - ( 'Y l y For example, L, = + + ()/ + ()(l)/ = 8. Also, the equatio whose roots a r e the th roots of x - x - = is x - L x + (-) =. (Essetially, equatio (A. ), a s w e l l a s (A. ), (B. ), (B ), appears i [ ] by Buschma. ) A alterate expressio for L is obtaied by expressig r + r i t e r m s of p ad A, where A = \/p + rq e Now, T i _LA\/-? o - /. ~ l. (-l) ~. (-l)(-) -, r x = L( +A )/J = (p + p A + - ^ T " A + / 'p + r, \ / o i o~ / _ = l.. (-l) -,(-l)(-) -, r = [( -A )/J ( - A + " ^ T ~ - A +-^r + which, for p = q =, A = v, implies, o addig, (A. ) L = ~ (l - (-l)/i + (~ l)(-)(-)/! +... ) [/] _i,, T ^;.)? - l x r Z i= B. THE DIFFERENCES OF THE th OWERS Now let us tur to the equatio whose roots a r e r ad - r where r ad r a r e the roots of the primitive equatio x -px -q = Ct Sice r l " r = ( p + ^ p + r q ) / / " ( p " V pl + q ) / / = ^ p + q = A ' the equatio whose roots a r e r, ad ~r is x - y p + rq x - r, r =, ad f(x) = x - A x + q =. Similarly, *\- x \ = O + / + ( )/ ] - [( - V/ + <>/ ] =

7 Q U A D R A T I C E Q U A T I O N A p r i l C o t i u i g i t h i s m a e r, a d a s s e m b l i g r e s u l t s, we h a v e T A B L E r l " r A. A (p + q ) A ( p + pq)a ( p + p q + q ) A ( p + p q + p q ) A ( p + p q + p q + q )A., -. _. - ^ ( - ) ( - ) - (p + (-)p q + - i JJ ' p q +..., ( - m - l ) ( - m - ),., ( - m ) - m - l m.. + _ Li ' i L p q ) A mi The sequece for r - r? as give i Table eds, for a give, o arrivig at t e r m z e r o. Writig the umerical coefficiets from Table i the form of a a r r a y of m colums ad rows, ad lettig C (m, ) deote the coefficiet i the mth colum ad th row, TABLE : C (m, ) A m % C A (m, ) L L L L L L L 8 L L

9 8 QUADRATIC EQUATION April (B.) F = + (-) + (-)(-)/! + (-)(-)(-)/! +... j=o the sum of t e r m s i risig diagoals of a s c a l ' s triagle. Also, the quadratic equatio whose roots a r e a ad -(, th powers of the roots of the quadratic equatio x - x - = with roots a = ( + / ) / ad ( = ( - / ) /, is x - /" F x - =. Expressig r, - r? i t e r m s of oly p ad A ad assemblig t e r m s as we did for r, + r i (A. ), we a r r i v e at a alterate exp r e s s i o for the th Fiboacci umber, (B. ) F = ~ ( -I- (-l)(-)/! + (- l)(-)(~)(-)/! +... [(-l)/] R < X " ( i + l J i= REFERENCES le S. L Basi, "A Note o Warig's Formula for Sums of Like owers of Roots", 9-. Fiboacci Quarterly :, April, 9, pp.. L. E. Dickso, F i r s t Course i the Theory of Equatios, Joh Wiley, 9. ~. R. G. Buschma, "Fiboacci Numbers, Chebyshev olyomials, Geeralizatios ad Differece Equatios", Fiboacci Quarterly :, D e c, 9, pp aul F. Byrd, "Expasio of Aalytic Fuctios i olyomials Associated with Fiboacci Numbers", Fiboacci Quarterly, :, Feb. 9, pp aul F. Byrd, "Expasio of Aalytic Fuctios i T e r m s Ivolvig Lucas Numbers or Similar Number Sequeces", Fiboacci Quarterly :, April, 9, pp. -. XXXXXXXXXXXXXXX

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