Recurrence Relations

1
Recurrence Relations

• the recursive formula for a recursively defined
  sequence is called a recurrence relation
• initial condition of the sequence must be defined
• Fibonacci sequence f1f21, fn fn-1 fn-2
• One technique for finding an explicit formula for
  the recursively defined sequence is backtracking,
  by substituting the previous terms till the
  initial condition
• a recurrence relation is a linear homogeneous
  relation of degree k if
• an r1an-1 r2an-2 rkan-k with the ris
  constants and rk?0
• its characteristic equation is
• xk r1xk-1 r2xk-2 rk

2
Recurrence Relations

• For degree 2, the following theorem gives a
  solution for finding the explicit formula for
  recurrence relation.
• Theorem
• If the characteristic equation x2-r1x-r20 of the
  recurrence relation an r1an-1 r2an-2 has two
  distinct roots, s1 and s2, then an us1n vs2n,
  where u and v depend on the initial conditions,
  is the explicit formula for the sequence.
• If the characteristic equation x2-r1x-r20 has a
  single root s, the explicit formula is an usn
  vnsn, where u and v depend on the initial
  conditions.