HW Wk 10: Recurrence Relations

Let an be the number of ways to climb n stairs if a person climbing the stairs can take one stair or two stairs at a time. Identify the number of ways the person who can take one stair or two stairs at a time can climb a flight of eight stairs.

34

There are 345 students at a college who have taken a course in calculus, 214 who have taken a course in discrete mathematics, and 190 who have taken courses in both calculus and discrete mathematics. How many students have taken a course in either calculus or discrete mathematics?

369 Let C be the set of students who have taken a course in calculus; thus, |C| = 345. Let D be the set of students who have taken a course in discrete mathematics; thus, |D| = 214. Then, C ∩ Drepresents the set of students who have taken courses in both calculus and discrete mathematics; thus, |C ∩ D| = 190. Using the principle of inclusion-exclusion, we have |C ∪ D| = |C| + |D| - |C ∩ D|. By substituting the values of |C|, |D|, and |C ∩ D|, we get |C ∪ D| = 345 + 214 - 190 = 369

Find the number of ways to distribute six different toys to three different children such that each child gets at least one toy. The number of ways to distribute six different toys to three different children such that each child gets at least one toy is

540

Find the number of positive integers not exceeding 100 that are either odd or the square of an integer.

55 Clearly, there are 50 odd positive integers not exceeding 100 (half of the 100 numbers are odd), and there are 10 squares. Furthermore, half of these squares are odd. Thus, we compute the cardinality of the set in the question to be 50 + 10 - 5 = 55.

Find the number of positive integers not exceeding 1020 that are not divisible by 3, 17, or 35.

621 |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C| ∣∣A∪B∪C∣∣=⌊1020/3⌋+⌊1020/17⌋+⌊1020/35⌋−⌊1020/3·17⌋−⌊1020/3·35⌋−⌊1020/17·35⌋+⌊1020/3·17·35⌋|A∪B∪C|=10203+102017+102035-10203·17-10203·35-102017·35+10203·17·35 |A ∪ B ∪ C| = 340 + 60 + 29 - 20 - 9 - 1 + 0 = 399 The number of positive integers not exceeding 1020 that are divisible by at least one of 3, 17, and 35 is 399; so, the number of positive integers not exceeding 1020 that are not divisible by 3, 17, or 35 numbers is 1020 - 399 = 621.

Find the number of positive integers not exceeding 112 that are not divisible by 5 or by 7.

77 To find the number of positive integers not exceeding 112 that are not divisible by 5 or by 7, we will subtract from 112 the number of positive integers that are divisible. 112-(112/5)-(112/7)+(112/5·7) = 112 - 22 - 16 + 3 = 112 - 35 = 77 positive integers that are not divisible by 5 or by 7. Find the number of positive integers not exceeding 104 that are not divisible by 5 or by 7.

Let an be the number of bit strings of length n that do not contain three consecutive 0s. Identify the number of bit strings of length seven that do not contain three consecutive 0s.

81

Let an be the number of ways to climb n stairs if a person climbing the stairs can take one stair or two stairs at a time. Identify the initial condition for the recurrence relation in the previous question. (You must provide an answer before moving to the next part.)

a0 = 1 and a1 = 1

Let an be the number of bit strings of length n that do not contain three consecutive 0s. Identify the initial conditions for the recurrence relation in the previous question. (You must provide an answer before moving to the next part.)

a0 = 1, a1 = 2, and a2 = 4

Consider the nonhomogeneous linear recurrence relation an = 2an − 1 + 2n. Identify the solution of the given recurrence relation with a0 = 2.

an = (n + 2)2n

olve these recurrence relations together with the initial conditions given. Identify the solution of the recurrence relation an = 6an − 1 - 8an − 2 for n ≥ 2 together with the initial conditions a0 = 4 and a1 = 10.

an = 3 · 2n + 4n

Let an be the number of bit strings of length n that do not contain three consecutive 0s. Identify a recurrence relation for an. (You must provide an answer before moving to the next part.)

an = an - 1 + an - 2 + an - 3 for n ≥ 3

Let an be the number of ways to climb n stairs if a person climbing the stairs can take one stair or two stairs at a time. Identify a recurrence relation for an. (You must provide an answer before moving to the next part.)

an = an −1 + an − 2 for n ≥ 2

Consider the nonhomogeneous linear recurrence relation an = 2an − 1 + 2n. Identify the set of all solutions of the given recurrence relation using the theorem given below. If {a(p)n} is a particular solution of the nonhomogeneous linear recurrence relation with constant coefficients an = c1an − 1 + c2an − 2 +· · ·+ckan − k + F(n), then every solution of the form {a(p)n + a(h)n}, where {a(h)n} is a solution of the associated homogeneous recurrence relation an = c1an − 1 + c2an − 2 +· · ·+ckan − k.

an = α(2)n + n(2)n