Question

Exercise 6.6. Let the inductive set be equal to all natural numbers, N. Prove the following propositions. (a) ∀n, 2n ≥ 1 + n.

(b) ∀n, 4n − 1 is divisible by 3.

(c) ∀n, 3n ≥ 1 + 2 n.

(d) ∀n, 21 + 2 2 + ⋯ + 2 n = 2 n+1 − 2.

Answer #1

Prove the following using induction:
(a) For all natural numbers n>2, 2n>2n+1
(b) For all positive integersn,
1^3+3^3+5^3+···+(2^n−1)^3=n^2(2n^2−1)
(c) For all positive natural numbers n,5/4·8^n+3^(3n−1) is
divisible by 19

prove that 2^2n-1 is divisible by 3 for all natural numbers n ..
please show in detail trying to learn.

Let N2K be the set of the first 2k natural numbers.
Prove that if we choose k + 1 numbers out of these 2k, there is at
least one pair of numbers a, b for which a is divisible by b.

Prove by induction on n that 13 | 2^4n+2 + 3^n+2 for all natural
numbers n.

Let S be the set {(-1)^n +1 - (1/n): all n are natural
numbers}.
1. find the infimum and the supremum of S, and prove that these
are indeed the infimum and supremum.
2. find all the boundary points of the set S. Prove that each of
these numbers is a boundary point.
3. Is the set S closed? Compact? give reasons.
4. Complete the sentence: Any nonempty compact set has a....

Show by induction that 1+3+5+...+(2n-1) = n^2 for all n in the
set of Natural Numbers

Let A =
3
1
0
2
Prove An =
3n
3n-2n
0
2n
for all n ∈ N

Let S = {0,1,2,3,4,...}, A = the set of natural numbers
divisible by 2, and B = the set of numbers divisible by 5. What is
the set A intersection B? What is the set A union B? Please show
your work.

. Prove that 2^(2n-1) + 3^(2n-1) is divisible by 5 for
every natural number n.

Let A be the set of all natural numbers less than 100.
How many subsets with three elements does set A have such that the
sum of the elements in the subset must be divisible by 3?

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 7 minutes ago

asked 11 minutes ago

asked 18 minutes ago

asked 18 minutes ago

asked 21 minutes ago

asked 38 minutes ago

asked 43 minutes ago

asked 58 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago