Question

Use mathematical induction to prove that 3^{n} ≥
n2^{n} for n ≥ 0. (Note: dealing with the base case may
require some thought.

Please explain the inductive step in detail.

Answer #1

Use Mathematical Induction to prove that 3n < n! if n is an
integer greater than 6.

Please note n's are superscripted.
(a) Use mathematical induction to prove that 2n+1 +
3n+1 ≤ 2 · 4n for all integers n ≥ 3.
(b) Let f(n) = 2n+1 + 3n+1 and g(n) =
4n. Using the inequality from part (a) prove that f(n) =
O(g(n)). You need to give a rigorous proof derived directly from
the definition of O-notation, without using any theorems from
class. (First, give a complete statement of the definition. Next,
show how f(n) =...

Use Mathematical Induction to prove that for any odd integer n
>= 1, 4 divides 3n+1.

“For every nonnegative integer n, (8n – 3n) is a multiple of
5.”
(That is, “For every n≥0, (8n – 3n) = 5m, for some m∈Z.” )
State what must be proved in the basis
step.
Prove the basis step.
State the conditional expression that must be proven in the
inductive step.
State what is assumed true in the inductive hypothesis.
For this problem, you do not have to complete the inductive step
proof. However, assuming the inductive step 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...

Use mathematical induction to prove the solution of problem T(n)
= 9T(n/3) + n, T(n) = _____________________________. is correct
(Only prove the big-O part of the result. Hint: Consider
strengthening your inductive hypothesis if failed in your first
try.)

Using mathematical induction show that
3n < n!, when n > 6

Prove by induction that 3^n ≥ 5n+10 for all n ≥ 3.
I get past the base case but confused on the inductive step.

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

(10) Use mathematical induction to prove that
7n – 2n is divisible by 5
for all n >= 0.

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