Question

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

Answer #1

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

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

(a) use mathematical induction to show that 1 + 3 +.....+(2n +
1) = (n + 1)^2 for all n e N,n>1.(b) n<2^n for all n,n is
greater or equels to 1

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) =...

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

Prove using mathematical induction that
20 + 21 + ... + 2n =
2n+1 - 1 whenever n is a nonnegative
integer.

Use mathematical induction to prove that
12+22+32+42+52+...+(n-1)2+n2=
n(n+1)(2n+1)/6. (First state which of the 3 versions of induction:
WOP, Ordinary or Strong, you plan to use.)
proof: Answer goes here.

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 show that ?! ≥ 3? + 5? for all integers ?
≥ 7.

Use the Strong Principle of Mathematical Induction to prove that
for each integer n ≥28, there are nonnegative integers x and y such
that n= 5x+ 8y

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