Last Update: 22 November 2010
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…Previous lecture
& a recurrence relation that defines the sequence in terms of its previous O/P:
recurrence relation:
a_{0} | 0 | 0 | 1 | 1 | 1 | 2 | 2 |
a_{1} | 0 | 1 | 0 | 1 | 2 | 1 | 2 |
a_{2} | 0 | 3 | –2 | 1 | 4 | –1 | 2 |
a_{3} | 0 | 7 | –6 | 1 | 8 | –5 | 2 |
a_{4} | 0 | 15 | –14 | 1 | 16 | –13 | 2 |
a_{5} | 0 | … | –30 | 1 | … | … | … |
… | … | … | … | … | … | … | … |
a_{n} | 0 | 2^{n}–1 | 2–2^{n} | 1 | 2^{n} | 3–2^{n} | 2 |
But the actual interest depends on their initial deposit!
where c_{1}, c_{2} ∈ R & c_{2} ≠ 0
is a linear homegeneous recurrence relation of degree 2
and can have differing intial conditions,
yielding different Fibonacci
sequences:
we have:
Let C_{0}, C_{1} ∈ N be constants.
Let a_{0} = C_{0} and a_{1} = C_{1} be the initial conditions of a recurrence relation.
Let c_{1}, c_{2} ∈ R be such that a_{n} = c_{1}a_{n–1} + c_{2}a_{n–2} is the recurrence relation.
Let r_{1} ≠ r_{2} be 2 distinct roots of the "characteristic equation"
of the recurrence relation. Then: