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Unformatted text preview: 18.06 Professor Johnson FINAL EXAM May 19, 2009 Grading 1 2 3 4 5 6 Total: Your PRINTED name is: Please circle your recitation: (R01) M2 2314 Qian Lin (R02) M3 2314 Qian Lin (R03) T11 2251 Martina Balagovic (R04) T11 2229 Inna Zakharevich (R05) T12 2251 Martina Balagovic (R06) T12 2090 Ben Harris (R07) T1 2284 Roman Bezrukavnikov (R08) T1 2310 Nick Rozenblyum (R09) T2 2284 Roman Bezrukavnikov I AGREE NOT TO DISCUSS THE CONTENTS OF THIS EXAM WITH ANY STUDENTS WHO HAVE NOT YET TAKEN IT UNTIL AFTER WEDNES DAY, MAY 20. (YOUR SIGNATURE) 1 (18 pts.) A sequence of numbers f ,f 1 ,f 2 ,... is defined by the recurrence f k +2 = 3 f k +1 f k , with starting values f = 1, f 1 = 1. (Thus, the first few terms in the sequence are 1 , 1 , 2 , 5 , 13 , 34 , 89 ,... .) (a) Defining u k = f k +1 f k , reexpress the above recurrence as u k +1 = A u k , and give the matrix A . (b) Find the eigenvalues of A , and use these to predict what the ratio f k +1 /f k of successive terms in the sequence will approach for large k . (c) The sequence above starts with f = f 1 = 1, and  f k  grows rapidly with k . Keep f = 1, but give a different value of f 1 that will make the sequence (with the...
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This note was uploaded on 01/14/2011 for the course EECS 18.06 taught by Professor Strang during the Spring '05 term at University of Michigan.
 Spring '05
 Strang

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