Question

For any odd integer n, show that 3 divides 2^{n}+1.

That is 2 to the nth power, not 2 times n

Answer #1

Using Mathematical Induction

2^n +1 is divisible by 3 for all odd integers n >= 1

Base Case:

Let n = 1,

LHS = 2^1 + 1 = 3

3 is divisible by 3.

The statement is true for n = 1.

Let the statement be true for k'th odd integer = (2k-1)

That is, 2^(2k-1) + 1 is divisible by 3.

=> 2^(2k-1) + 1 = 3p

=> 2^(2k-1) = 3p - 1

Consider the statement for (k+1)'th odd integer = (2k+1)

LHS = 2^(2k+1) + 1 = 2^(2k-1 + 2) + 1 = 2^(2k-1) * 2^2 + 1 = (3p-1)*4 + 1 = 3p*4 + 3 = 3*(4p + 1)

3*(4p + 1) is divisible by 3.

=> 2^(2k+1) + 1 is divisible by 3

The statement is true for (k+1)'th odd integer,

The statement is true for all odd integers n >=1

Consider the following statement: if n is an integer,
then 3 divides n3 + 2n.
(a) Prove the statement using cases.
(b) Prove the statement for all n ≥ 0 using
induction.

If n is an odd integer, prove that 12 divides
n2+(n+2)2+(n+4)2+1. Please provide
full solution!

Used induction to proof that 1 + 2 + 3 + ... + 2n = n(2n+1) when
n is a positive integer.

Show that, for any integer n ≥ 2, (n + 1)n − 1 is
divisible by n2 . (Hint: Use the Binomial Theorem.)

Prove that for each positive integer n, (n+1)(n+2)...(2n) is
divisible by 2^n

3.a) Let n be an integer. Prove that if n is odd, then
(n^2) is also odd.
3.b) Let x and y be integers. Prove that if x is even and y is
divisible by 3, then the product xy is divisible by 6.
3.c) Let a and b be real numbers. Prove that if 0 < b < a,
then (a^2) − ab > 0.

True Or False
1. If nn is odd and the square root of nn is a natural number
then the square root of nn is odd.
2. The square of any even integer is even
3. The substraction of 2 rational numbers is rational.
4. If nn is an odd integer, then n2+nn2+n is even.
5. If a divides b and a divides c then a divides bc.
6. For all real numbers a and b, if a^3=b^3 then a=b.

6. Consider the
statment. Let n be an integer. n is odd if and
only if 5n + 7 is even.
(a) Prove the forward implication of this statement.
(b) Prove the backwards implication of this statement.
7. Prove the following statement. Let a,b, and
c be integers. If a divides bc and
gcd(a,b) = 1, then a divides c.

Show that, for any positive integer n, n lines ”in general
position” (i.e. no two of
them are parallel, no three of them pass through the same point)
in the plane R2
divide the plane into exactly n2+n+2 regions. (Hint: Use the
fact that an nth line 2
will cut all n − 1 lines, and thereby create n new regions.)

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...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 4 minutes ago

asked 24 minutes ago

asked 33 minutes ago

asked 42 minutes ago

asked 46 minutes ago

asked 48 minutes ago

asked 55 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago