Question

help please!

If do=1 92=4 anty-an tant. Find the closed form of an.

Answer 1

Ques: > with autant tuz1 ... Cat?:? Given a = 1 and a₂=4 to anti- ant au- With q=1 and 9=4 We get, - ₂ = 9,+ 7 4=+1 6,19-1 and 0,=3) Nolor, Let F(x)= 5 and ... 20 ALSO, anti ant and tus1 which is equiralent to i * Oy = naj + anaz #na2 -. multiplying @ by x² on both sides, we get an...x + an-2 x tuzz ns2 anura + L-2 9n-2. X + x 92 un no (,) + +(364) = 2 (F(x) —%) + x2 ( F(x)) 2 * = x (F12)-1) + x* (ples) na

→ = Fla) - 9-9,2 = x( Flve) - )) + red flse) F(2)– 1 - 3 = x Flx) - x + xe? F(x) se? Else) x Flve) - F13) = -32-1 tz 7 (Flve)) (2+x-1) = -1-2x => Fle) = -(1+22) 21? + -! Now, we would like to factorise x²+x-1 x² + x -l=0 2 2 5 (x+ 55 *') AND (x- 1631) gre factors .Fla) = ) (- ) ) (2x f(3) = - (xt J 5

Let us denote by a= ust! b = 5-] 2 Then, ) F()= -(206+ (uta) (x-b) He shall decompose this fraction now So, het - (2x+1) = A (a +9) (-6) 12 +9) C2-b) - (2x +1)= A (-5)B(xfa) X16 Putting x=bi we get - J5-171) B (b+a)= -(2671) => B( 15-1495 +)= = B( 15 )= -V5 7 B=-1 AND, x=-a, putting 1. ure ger; +22 -1 = A(-9-b) =-1(a+b) = 29-1. 7- A (V5)= 15+)-1 - V5 =) Az-1

aeger, + (-) | F(x) - (ax+1) | (7) (-) t a 6 movi Sue We get LA ata | ㅋ 4 () C) () Similarly, 다 . > - - = 1. 5 1 2 o F(x) 네 + Fax: -l 저다 | + -5 다 . 6가 캐 +)

7 reso corte tra]* → FO) = bnt] Corro This, I we ano an= (-jht1 get anti tanto ntb We aga Since, a= √5 + 1 and, bo= 15 - 1 Cena 2015november in the -srs] get Ans ) verifications - a=1 for n=o, And, + [ + D 1.2 ( 1534445*') - 4+1 - valid for n=o. o for n=1; 9=3. AND, 25 + I 2 + 155-15 , 7 = 4.5 5+1-2 55+ 5+1 +2 15 7 L (5-1)² = 4(2- -> valid for n=1.