Prove by induction that 14^n + 12^n −5^n is divisible by 7 for all n >0

Expert Solution

The quantity Q will be an integer for any value of k. We can say anything is divisible by 7 if I can take out a common factor of 7 from the expression.

Please consider giving a thumbs up if you find the solution helpful.

Thank you & All the best!

genius_generous answered 1 year ago

Use induction to prove that 8^n - 3^n is divisible by 5 for all integers n>=1.

12 pts) Use Mathematical Induction to prove that an=n3+5n is divisible by 6 when ever n≥0....

12 pts) Use Mathematical Induction to prove that an=n3+5n is divisible by 6 when ever n≥0. You may explicitly use without proof the fact that the product n(n+1) of consecutive integers n and n+1 is always even, that is, you must state where you use this fact in your proof.Write in complete sentences since this is an induction proof and not just a calculation. Hint:Look up Pascal’s triangle. (a) Verify the initial case n= 0. (b) State the induction hypothesis....

a) Prove by induction that if a product of n polynomials is divisible by an irreducible...

a) Prove by induction that if a product of n polynomials is divisible by an irreducible polynomial p(x) then at least one of them is divisible by p(x). You can assume without a proof that this fact is true for two polynomials. b) Give an example of three polynomials a(x), b(x) and c(x), such that c(x) divides a(x) ·b(x), but c(x) does not divide neither a(x) nor b(x).

Prove that 3^n + 7^(n−1) ≡ 4 (mod 12) for all n ∈ N+.

5. Without using the method of mathematical induction, prove that 5^n − 3^n + 2n is...

5. Without using the method of mathematical induction, prove that 5^n − 3^n + 2n is divisible by 4 for all natural n.

Prove these scenarios by mathematical induction: (1) Prove n2 < 2n for all integers n>4 (2)...

Prove these scenarios by mathematical induction: (1) Prove n2 < 2n for all integers n>4 (2) Prove that a finite set with n elements has 2n subsets (3) Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps

Question

Prove by induction that 14^n + 12^n −5^n is divisible by 7 for all n >0

Solutions

Expert Solution

Related Solutions

Use induction to prove that 8^n - 3^n is divisible by 5 for all integers n>=1.

12 pts) Use Mathematical Induction to prove that an=n3+5n is divisible by 6 when ever n≥0....

a) Prove by induction that if a product of n polynomials is divisible by an irreducible...

Prove that 3^n + 7^(n−1) ≡ 4 (mod 12) for all n ∈ N+.

5. Without using the method of mathematical induction, prove that 5^n − 3^n + 2n is...

Prove these scenarios by mathematical induction: (1) Prove n2 < 2n for all integers n>4 (2)...

Prove by strong mathematical induction that any integer greater than 1 is divisible by a prime...

Use a mathematical induction for Prove a^(2n-1) + b^(2n-1) is divisible by a + b, for...

Use double induction to prove that (m+ 1)^n> mn for all positive integers m; n

If you prove by strong induction a statement of the form ∀ n ≥ 1P(n), the...