Question

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

Answer #1

Prove that for n>=1, (2n-1)^2-1 is divisible by 8.

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

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

Use mathematical induction to prove that for each integer n ≥ 4,
5n ≥ 2 2n+1 + 100.

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

Discrete math
Use mathematical induction to prove that n(n+5) is divisible by
2 for any positive integer n.

Suppose for each positive integer n, an is an integer such that
a1 = 1 and ak = 2ak−1 + 1 for each integer k ≥ 2. Guess a simple
expression involving n that evaluates an for each positive integer
n. Prove that your guess works for each n ≥ 1.
Suppose for each positive integer n, an is an integer such that
a1 = 7 and ak = 2ak−1 + 1 for each integer k ≥ 2. Guess a...

Prove that 2n < n! for every integer n ≥ 4.

Let n be an integer, with n ≥ 2. Prove by contradiction that if
n is not a prime number, then n is divisible by an integer x with 1
< x ≤√n.
[Note: An integer m is divisible by another integer n if there
exists a third integer k such that m = nk. This is just a formal
way of saying that m is divisible by n if m n is an integer.]

Prove by induction that k ^(2) − 1 is divisible by 8 for every
positive odd integer k.

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