Transcription of THE FIBONACCI NUMBERS
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THE FIBONACCI NUMBERS
\Fibonaccinumbers"is usedto describe theseriesof numbersgener-atedby thepattern1;1;2;3;5;8;13;21;34;55;89;144 :::,whereeach number in thesequenceis given by thesumof theprevioustwo given byu1= 1,u2= 1 andtherecursive formulaun=un 1+un 2; n > fromthefamous\rabbitproblem"of 1228,theFibonaccinumberswereoriginallyus edto represent thenumber of pairsof rabbitsbornof onepairin a assumethata pairof rabbitsis introducedinto acertainplacein the rstmonth of rabbitswillproduceonepairof o springeverymonth,andeverypairof rabbitswillbeginto reproduceexactlytwo dies,andeverypairof ,in the rstmonth,we have onlythe rstpairof ,in thesecondmonth,we againhave onlyourinitialpairof ,by thethirdmonth,thepairwillgive birthto anotherpairof rabbits,andtherewillnowbe two ,we ndthatin monthfourwe willhave 3 pairs,then5 pairsin month ve, then8,13,21,34.
Until now, we have primarily been using term-by-term addition to nd formulas for the sums of Fibonacci numbers. We will now use the method of induction to prove the following important formula. Lemma 6. Another Important Formula un+m = un 1um +unum+1: Proof. We will now begin this proof by induction on m. For m = 1, un+1 = un 1 +un = un 1u1 +unu2;
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