Question

linear algebra please help

3.00 The purpose of this exercise is to find an explicit formula for the entries of the following recursively defined sequence: fn+1 fr +25n-1 if n 1 fi = fo=1. (a) Write down the first 6 entries of this sequence. (b) Now, introduce the vectors futi (1) fo for n € {0,1,2...}. Find a matrix A € R2x2 such that F = AF-1 and explain why = A. (c) Consequently, by diagonalizing A, find the explicit formula for fr

Answer 1

defined recursively as inzi (1) la/ The first 6 entries Griven sequence is turl= fut2 tnt fiato la of the sequence are given by for1, fial, fat, +260=3, t32 f2+2f = 3+2=5, fqz fz +262 = 5+2*3 21) ts=ta +242 - 11+2x5= 162 F = ( fat! ), n8 50,1,2,-) 21 Use the vectors / by all tu 2 tu turi o A fnl, where A2 -0%) and similary, 2 Fn = A (AF2) z A² Fn-z Fu = A A² Fur j -(2) (C) 6 2 2. A" Form 2 A" Now, A (-)-(13)()-()=-1)() and Al?)= (?)(?) (1) 221 7(), (?) are eigenvectors of A eigenvalues 1/24, 12 = 2 P2 (43) P". A P =D (62) with com. let, then Ps į C ;) and Dz 2 A = PD pt : A2 Poup Fu P.php ( Tap the 6-733770" 2.1 GH(!)

( 2b+i), 2 2 (3) ()() 이 (2) 2. 1, 2 7 (1) +) Listipsd 25 3 1 3. nt2 J (2 + (4) (A))) 3 ㅗ + (AM) fna (2**' + (4)") 1 120 its tsh 2