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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 2-314 Qian Lin (R02) M3 2-314 Qian Lin (R03) T11 2-251 Martina Balagovic (R04) T11 2-229 Inna Zakharevich (R05) T12 2-251 Martina Balagovic (R06) T12 2-090 Ben Harris (R07) T1 2-284 Roman Bezrukavnikov (R08) T1 2-310 Nick Rozenblyum (R09) T2 2-284 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 , re-express 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