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. Show that the sequence {fn} defined explicitly by fn= 2(-4)n +3 is a
solution of the recurrence relation fn = -3 fn-1+4 fn-2.
3. Suppose that the discrete function f is defined recursively by:
f(0) =2 and
f(n+1) = 2f(n) +3
Then find f(1), f(2), f(3), f(4) and f(5)
4. Find a recursive definition for the following sequence
a. an = {2,4,8,16,32, …}, where n 1,
b. an ={3,6,9,12,15, … ) , where n 1,
5. Give a recursive definition of the sequence {fn} , if
(a) fn =5n (b) fn = 2n+1
(c) fn+1 = 10n (d) fn = 5.
6. Find the characteristic equation of each of the following recurrence
relations
a. fn = 3fn-1 -2fn-2, n > 2 with initial conditions: f1=3 and f2 = 7
b. yn= 6yn-1- 9yn-2, n>1, y0= 5 and y1= 3
7. Solve the following recurrence relations.
(a) an =5an-1 +6an-2, n 2 with a0 =1, a1= 3.
(b) 2an+2 –11an+1+ 5an = 0, n 0 with a0= 2, a1= -8.
(c) 3an+1= 2an+ an-1, n 1, a0 =7, a1 = 3
(d) an – 6an-1+ 9an-2 = 0, n 2, a0 =5, a1 =12.
Show that the explicit sequence {an} where an= 2n+1
-1 for n > or = to 1 is a
solution of the recurrence relation: an= 3an-1 – 2an-2 , n >or = to 3
6 divides n^3 - n for n>=2
Prove by mathematical induction: 6 divides n^3 - n for n>=2
Let A = {a, b, {b}, {b, e}, e} and B = {b, {b, e}, e, f}.
A ⋂ B = {b, {b, e}, e}
a. false
b. true
Construct a combinatorial circuit using inverters, OR gates, and AND gates that produces the output
￢((a ∨ b) ∧ (a ∨ c)) ∨ (b ∨ c) from input bits a, b and c.
Determine whether each statement is trautology, contradiction or contingency
1. (P v~q) -> (q^~ r)
2. ~[r -> (q ^ r)]
3. (P v q) -> (q ^ r)
4. q -> ~ (q ^ r)
5. ~ [ t v (s ^ r)]
From a committee of 10 people,
1. In how many ways can we choose a chairperson, a vice-chairperson, and a secretary, assuming that one person cannot hold more than one position?
2. In how many ways can we choose a subcommittee of 3 people?
3. Find the number of combinations of 13 objects taken 8 at a time.
4. How many 5-card hands will have 3 aces and 2 kings?
5. Serial numbers for a product are to be made using 2 letters followed by 3 numbers. If the letters are to be taken from the first 8 letters of the alphabet with no repeats and the numbers are taken from the 10 digits (0-9) with no repeats, how many serial numbers are possible?
6. A company has 7 senior and 5 junior officers. An ad hoc legislative committee is to be formed. In how many ways can a 4-officer committee be formed so that it is composed of
a. Any 4 officers?
b. 4 senior officers?
c. 3 senior officers and 1 junior officer?
d. 2 senior officers and 2 junior officers?
e. At least 2 senior officers?
Find the value of a2 for the recurrence relation an=17an-1+30n, where a0=3
Find the value of a3 for the recurrence relation an=17an-1+30n, where a0=3
Find the value of a1 for the recurrence relation an=17an-1+30n, where a0=3
What would be the hypothesis of the mathematical induction for x(x + 1) < x! , where x ≥ 7?
If P(k) = k2(k + 2)(k – 1) is true, then what is P (k + 1)?
A. Determine if 1781 is divisible by 3, 6, 7, 8, and 9. (5 items x 2 points)
B. Determine if each of the following numbers is a prime or composite.
6. 828
7. 1666
8. 1781
9. 1125
10. 107
C. Find the greatest common divisor of each of the following pairs of integers.
11. 60 and 100
12. 45 and 33
13. 34 and 58
14. 77 and 128
15. 98 and 273
D. Find the least common multiple of each of the following pairs of integers.
16. 72 and 108
17. 175 and 245
18. 150 and 70
19. 32 and 27
20. 540 and 504
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