Fall 2004 - Popescu's Class - Exam 2

# Fall 2004 - Popescu's Class - Exam 2 - CD SOLUTmNS Exam 2...

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Unformatted text preview: CD SOLUTmNS Exam 2, Mathematics 109 Name: Dr. Cristian D. Popescu Student ID: November 15, 2004 - ' Section Number: Note: This exam conSists of 3 problems worth a total of 100 points. You will ‘ not receive credit unless you ShOW all your work. No books, calculators, notes or tables are permitted. ' I. (40 pts.) For each natural number n 2 3, let An = (n, 71”] . (1) Show that, for all natural numbers n 2 3, we have n! E An. (2) Find U An and H An. Justify your answers. n23 n23 Recall: n!=1-2-3‘...‘n,forallnEN. We "HM %’ {X'kmﬂLlo‘ XY‘V‘V‘ 54V locket.an ’i‘b Show N CA U): “)3 45¢sz Wm) H ML Ne w“ 9:an YCM (mm 4 but“ QQ—HY‘H \5 .aJv-ro “m If ' w .;.\ L: «A 4 V0 -a-> LYN-O 4 n . 3 ﬂy (n+0 < T M m) N” am“ WW LMQM é 6H6“ m \ (i +I;”*'(/“ \ varq\ A aha-N) Cum) =3 Ch—HX‘ ._ w b” ==q Pin“) _)PCMW fem \$4 “)3 :3 CE, II. (30 pts.) Let {fn}neN be the Fibonacci sequence, deﬁned recursively by f1 2 f2 : 1, and fn = fn—I + fn_2, for all n 2 3. (1) Prove that, for each natural number n 2 1, fan is an even number. (2) Prove that, for each natural number n > 1, we have an equality w “GEN - Am“ M 1m» Mu . We mu tVCMO 9%“ ' m w my _.___.-.‘ '— L 2% 23% * ii“ = G) me GD III. (30 pts.) Give the explicit description of the solution set for the following Diophantine equation. 395-m—18O-y265. (a?) 3% = 2 \80 + ’21; \80 2 '\' :11 ’55 = 3v 'E. 4: <1 (Ru-(05M :KCA (3°55 H30) Smog 5\6§7 it} be) ww Col‘mls iiiuwm wine 5 :JJ 3, —~ :39. = -5C3ﬁ—2\ﬂ3 # ._. “53.33; i &—\\)-(-\?0 not t5 2 it; = Eco-w, + NO bum) : Eur‘m ...
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Fall 2004 - Popescu's Class - Exam 2 - CD SOLUTmNS Exam 2...

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