Prove by induction that: 1) x^n - 1 is divisible by x-1 2)2n < 3^n for...

Prove by induction that:

1) x^n - 1 is divisible by x-1

2)2n < 3^n for all natural numbers n

Expert Solution

Colby Messinger answered 6 months ago

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

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

Prove the following by induction: 2 + 4 + 6 + …+ 2n = n(n+1) for...

Prove the following by induction: 2 + 4 + 6 + …+ 2n = n(n+1) for all integers n Show all work

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

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

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).

By induction: 1. Prove that Σni=1(2i − 1) = n2 2. Prove thatΣni=1 i2 = n(n+1)(2n+1)...

By induction: 1. Prove that Σni=1(2i − 1) = n2 2. Prove thatΣni=1 i2 = n(n+1)(2n+1) / 6 .

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.

2. [6 marks] (Induction) Prove that 21 divides 4n+1 + 5 2n−1 whenever n is a...

2. [6 marks] (Induction) Prove that 21 divides 4n+1 + 5 2n−1 whenever n is a positive integer. HINT: 25 ≡ 4(mod 21)

Use induction to prove that 2 + 4 + 6 + ... + 2n = n2...

Use induction to prove that 2 + 4 + 6 + ... + 2n = n2 + n for n ≥ 1. Prove this theorem as it is given, i.e., don’t first simplify it algebraically to some other formula that you may recognize before starting the induction proof. I'd appreciate if you could label the steps you take, Thank you!

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

Question