Permutations and Combinations

Permutations and Combinations

Question

The Fibonacci sequence is defined by

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Find fraction numerator straight a subscript straight n plus 1 over denominator straight a subscript straight n end fraction, for n = 1, 2, 3, 4, 5.


Answer

The Fibonacci sequence is given by
                              straight a subscript 1 space equals space straight a subscript 2 space equals space 1
and                          straight a subscript straight n space equals space straight a subscript straight n minus 1 end subscript plus straight a subscript straight n minus 2 end subscript
Putting n =3 in (i), we get straight a subscript 3 space equals space straight a subscript 3 minus 1 end subscript space plus space straight a subscript 3 minus 2 end subscript space equals space straight a subscript 2 space plus space straight a subscript 1 space equals space 1 space plus space 1 space equals space 2
Putting n = 4 in (i), we get straight a subscript 4 space equals space straight a subscript 4 minus 1 end subscript space plus space straight a subscript 4 minus 2 end subscript space equals space straight a subscript 3 plus straight a subscript 2 space equals space 2 space plus space 1 space equals 3
Putting n = 5 in (i), we get straight a subscript 5 equals straight a subscript 5 minus 1 end subscript plus straight a subscript 5 minus 2 end subscript equals straight a subscript 4 plus straight a subscript 3 equals 3 plus 2 equals 5
Putting n = 6 in (i), we get straight a subscript 6 space equals space straight a subscript 6 minus 1 end subscript space plus space straight a subscript 6 minus 2 end subscript space equals space straight a subscript 5 plus straight a subscript 4 space equals space 5 space plus space 3 space equals 8
Now, let       straight t subscript straight n space equals space straight a subscript straight n plus 1 end subscript over straight a subscript straight n
                  straight t subscript 1 space equals space fraction numerator straight a subscript 1 plus 1 over denominator straight a subscript 1 end fraction space equals space fraction numerator straight a subscript 2 over denominator straight a subscript 1 space end fraction space equals space 1 over 1 space equals space 1 commastraight t subscript 2 space equals space fraction numerator straight a subscript 2 plus 1 over denominator straight a subscript 2 end fraction space equals space straight a subscript 3 over straight a subscript 2 space equals space 2 over 1 equals 2 comma space space space straight t subscript 3 space equals space fraction numerator straight a subscript 3 plus 1 over denominator straight a subscript 3 end fraction space equals space straight a subscript 4 over straight a subscript 3 equals space 3 over 2
                 straight t subscript 4 space equals space fraction numerator straight a subscript 4 plus 1 over denominator straight a subscript 4 end fraction space equals space straight a subscript 5 over straight a subscript 4 space equals space 5 over 3 comma space space straight t subscript 5 space equals space fraction numerator straight a subscript 5 plus 1 over denominator straight a subscript 5 end fraction space equals space straight a subscript 6 over straight a subscript 5 space equals space 8 over 5

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