\u062a\u0645\u0627\u0631\u064a\u0646 \u0644\u0644\u0645\u0631\u0627\u062c\u0639\u0647 \u0627\u0644\u0646\u0647\u0627\u0626\u064a\u0629 - Remarks read p415 binomial coefficient and identities carefully(Binomial Theorem is very important and

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Remarks: read p415 binomial coefficient and identities carefully (Binomial Theorem) is very important and all section 6.4 Concentrate on: 1- chapter 10 (graphs and representation by matrices). 2-chapter 9 relations 3-chapter 8, how to solve a recurrence relation 4-chapter 5, how to prove by mathematical induction. 5- chapter 1, how to prove that two statements are equivalent or to prove a statement is a tautology. Q1: which of the following are linear homogeneous recurrence relation with constant coefficient: a)a n =3a n-1 +4a n-2 +5a n-3 . b) a n =2na n-1 +a n-2 c) a n =a n-1 +2 d)a n =a n-1 2 +a n-2 e) a n =a n-1 +n sol: a) Yes, degree=3 degree= the largest- the smallest=n-(n-3). b) No, because there is “n” in the first term. c) Yes, degree=n-(n-1)=1. d) No, because there is 2 as apower in the first term e) No, because there is “n” in the second term.
Q 2: solve the following recurrence relation:a)an=an1,a0=2 α b)an=−4an14an2,a0=1,a1=1 n Q 3:List the ordered pairs in the relation R (a,b)fromA={0,1,2,3,4} to B={0,1,2,3} if and only if:
Q 4: Check reflexive, symmetric, transitive if R on definedon the set of all integers: a)R={(x , y):x ≠ y}b)R={(x , y):xy ≥1}c)R={(x , y):x=y+1x=y1} Solution: yRx . . . 1
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