Question

prove that n^3+2n=0(mod3) for all integers n.

Answer #1

Use Mathematical Induction to prove that 3 | (n^3 + 2n) for all
integers n = 0, 1, 2, ....

Prove that n − 1 and 2n − 1 are relatively prime, for all
integers n > 1.

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

Problem 1. Prove that for all positive integers n, we have 1 + 3
+ . . . + (2n − 1) = n ^2 .

For which positive integers n ≥ 1 does 2n > n2 hold? Prove
your claim by induction.

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 or disprove each of the following statements:
(a) For all integers a, a | 0.
(b) For all integers a, 0 | a.
(c) For all integers a, b, c, n, and m, if a | b and a | c, then
a | (bn+cm).

Prove that |U(n)| is even for all integers n ≥ 3. (use Lagrange’s
Theorem)

Prove that for all positive integers n,
(1^3) + (2^3) + ... + (n^3) = (1+2+...+n)^2

Prove that if n ≥ 2, then n! < S(2n, n) < (2n)!
S(2n,n) is referencing to Stirling Numbers

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 21 minutes ago

asked 24 minutes ago

asked 28 minutes ago

asked 47 minutes ago

asked 49 minutes ago

asked 58 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago