Question

This problem illustrates the importance of establishing the base case.

(a) Let P(n) be the following: 2 + 4 + 6 + ... + 2n = n(n + 1) + 2.

Assume that for some n ∈ N, P(n) is true. Using this as your hypothesis, show that P(n + 1) is true.

(b) Is P(n) true for all n?

Answer #1

Let P(n) be the statement that 12 + 22 +· · ·+n 2 = n(n+ 1)(2n+
1)/6 for the positive integer n. Prove that P(n) is true for n ≥
1.

Let a0 = 1, a1 = 2, a2 = 4, and an = an-1 + an-3 for n>=
3.
Let P(n) denote an an <= 2^n.
Prove that P(n) for n>= 0 using strong induction:
(a) (1 point) Show that P(0), P(1), and P(2) are true, which
completes
the base case.
(b) Inductive Step:
i. (1 point) What is your inductive hypothesis?
ii. (1 point) What are you trying to prove?
iii. (2 points) Complete the proof:

Let a0 = 1, a1 = 2, a2 = 4, and an = an-1 + an-3 for n>=
3.
Let P(n) denote an an <= 2^n.
Prove that P(n) for n>= 0 using strong induction:
(a) (1 point) Show that P(0), P(1), and P(2) are true, which
completes
the base case.
(b) Inductive Step:
i. (1 point) What is your inductive hypothesis?
ii. (1 point) What are you trying to prove?
iii. (2 points) Complete the proof:

Prove the following statement by mathematical induction. For
every integer n ≥ 0, 2n <(n + 2)!
Proof (by mathematical induction): Let P(n) be the inequality 2n
< (n + 2)!.
We will show that P(n) is true for every integer n ≥ 0. Show
that P(0) is true: Before simplifying, the left-hand side of P(0)
is _______ and the right-hand side is ______ . The fact that the
statement is true can be deduced from that fact that 20...

7. (Problem 3 on page 341 from Rosen) Let P(n) be the statement
that a postage of n cents can be formed using just 3-cent stamps
and 5-cent stamps. The parts of this exercise outline a strong
induction proof that P(n) is true for n ³ 8.
Show that the statements P(8), P(9), and P(lO) are true,
completing the basis step of the proof.
What is the inductive hypothesis of the proof?
What do you need to prove in the...

Definition: Let p be a prime and 0 < n then the p-exponent of
n, denoted ε(n, p) is the largest number k such that pk | n.
Note: for p does not divide n we have ε(n,p) = 0
Notation: Let n ∈ N+ we denote the set {p : p is prime and p |
n} by Pr(n). Observe that Pr(n) ⊆ {2, 3, . . . n} so that Pr(n) is
finite.
Problem: Let a, b be...

Let T(n) = 1 + 2 + ... + n be the n-th triangular number. For
example, t(1) = 1, t(2) = 3, t(3) = 6... T(n)= n(n+1)/ 2
a. Show that T(2n) = 3T(n) + T(n-1)
b. Show that T(1) + T(2) + T(n) = (n(n+1)(n+2))/6

Let P(n) be the statement that
13 + 23 + ... + n3 = (n(n+1)/2)2
Work with your group in the forum to prove P(n) is true for all
positive integers n

Assume S is a recursivey defined set, defined by the following
properties:
1 ∈ S
n ∈ S ---> 2n ∈ S
n ∈ S ---> 7n ∈ S
Use structural induction to prove that all members of S are
numbers of the form 2^a7^b, with a and b being non-negative
integers. Your proof must be concise.
Remember to avoid the following common mistakes on structural
induction proofs:
-trying to force structural Induction into linear Induction.
the inductive step is...

Problem 3.2
Let H ∈ Rn×n be symmetric and idempotent, hence a projection
matrix. Let x ∼ N(0,In). (a) Let σ > 0 be a positive number.
Find the distribution of σx. (b) Let u = Hx and v = (I −H)x and ﬁnd
the joint distribution of (u,v). 1 (c) Someone claims that u and v
are independent. Is that true? (d) Let µ ∈ Im(H). Show that Hµ = µ.
(e) Assume that 1 ∈ Im(H) and ﬁnd...

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