Question

Let A =

3 | 1 |

0 | 2 |

Prove An =

3^{n} |
3^{n}-2^{n } |

0 | 2^{n} |

for all n ∈ N

Answer #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 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

Show all the steps...
Prove by induction that 3n < 2n for all
n ≥ ______. (You should figure out what number goes in the
blank.)

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

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

can you please show all the steps thank you...
Prove by induction that 3n < 2n for all
n ≥ ______. (You should figure out what number goes in the
blank.)
I know that the answer is n>= 4, nut I need to write the
steps for induction

Let n be a positive odd integer, prove gcd(3n, 3n+16) = 1.

Exercise 6.6. Let the inductive set be equal to all natural
numbers, N. Prove the following propositions. (a) ∀n, 2n ≥ 1 +
n.
(b) ∀n, 4n − 1 is divisible by 3.
(c) ∀n, 3n ≥ 1 + 2 n.
(d) ∀n, 21 + 2 2 + ⋯ + 2 n = 2 n+1 − 2.

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

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

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