MAT 251 Discrete Mathematics

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MAT 251 Discrete Mathematics

• Recurrence Relations

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Section 7.2 Recurrence Relations

• Def A linear homogeneous recurrence relation of
  degree k with constant coefficients is a RR of
  the form
• an c1an-1 c2an-2 ckan-k ,
• where c1 , c2 , , ck are real numbers and ck
  is not zero.

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Section 7.2 Recurrence Relations

• Examples
• 1) fn fn-1 fn-2
• 2) an 5an-1 - 6an-2 c5an-5
• 3) an c4an-4

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Section 7.2 Recurrence Relations

• NON-Examples
• 1) an an-13 an-2
• 2) an an-1 an-2 70 3n
• 3) an (n-3)an-4
• 4) an an-3an-4

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Section 7.2 Recurrence Relations

• To solve a LHRR with constant
• coefficients means to find a sequence
• which satisfies it.
• We will only consider 2nd degree LHRR with
  constant coeffiecients and suggest that anyone
  interested to read the rest of the section for
  what happens with other types ?!

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Section 7.2 Recurrence Relations

• What is the idea!
• Given a 2nd degree LHRR with CC,
• an c1an-1 c2an-2,
  (1)
• We make a guess for a solution of the form an
  rn. This means
• that it satisfies (1), that is,
• rn c1 rn-1 c2 rn-2 (2)
• , If we divide both sides of (2) by rn-2 and
  set it equal to 0, we get
• r2 - c1 r - c2 0 (3)
• which we call this the characteristic
  equation of (1). This is a quadratic equation
  which we can solve and its roots are called the
  characteristic roots of (1).

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Section 7.2 Recurrence Relations

• THM 1 Given r2 - c1 r - c2 0 ()
• Then the sequence an is a solution of
• an c1an-1 c2an-2, (1)
• iff
• Case 1 If () has two distinct solutions, say r1
  and r2, then an a1 r1n a2 r2n
• where a1 and a2 are real numbers.
• Case 2 If () has a unique solution of
  multiplicity 2, say r0, then an a1 r0n a2
  nr0n
• where a1 and a2 are real numbers.

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Section 7.2 Recurrence Relations

• Find the solution of the RR
• 1) fn fn-1 fn-2 with f0 0 and f1 1.
• 2) an 5an-1 - 6an-2 with a0 1 and a1 0.
• 3) an 2an-1 - an-2 with a0 -1 and a1 1.
  
• 4) an 2an-1 - 2an-2 with a0 1 and a1
  2.

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Section 7.2 Recurrence Relations

• Lets do Page 471/8 together ?!

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Section 7.2 Recurrence Relations

• The notes have been created using K. H. Rosens
  Discrete Mathematics and Applications, 6th Edition