Discrete Math Proofs Please prove number one from Category C. Prove other number

Question

Discrete Math Proofs

Please prove number one from Category C. Prove other number one's also if possible.

Prove that the product of any two consecutive integers has either the form 3k or 3k +2, k elementof Z. If n is an integer, then n^3 - n - 1 is always odd. Prove that the number Squareroot 3 is irrational You may use the lemma "If 3 divides x^2, then 3 divides x" If a is a non-zero rational number and b is an irrational number, then 3a^2 b/7 is an irrational number. For all integers n greaterthanorequalto 0, 2^3n - 1 is divisible by 7 For all integers n greaterthanorequalto 1 and for r elementof R, r notequalto 0, r notequalto 1: sigma_i=0^n r^i = r^n+1 - 1/r - 1 A sequence a_0, a_1, a_2, ellipsis defined by a_k = 5 a_k-1 - 6 a_k-2 for all integers k greaterthanorequalto 2 Prove that for all integers n greaterthanorequalto 0, a_n = 5 middot 3^n - 3 middot 2^n For all sets A, B, and C, (A- B) Union (B-A) = (A Union B) - (A Intersection B)

Explanation / Answer

We have to prove that 23n - 1 is divisible by 7.

Now 23n-1

= (23)n-1

= 8n - 1

= (8 - 1)(8n-1 + 8n-2 + ... + 80

We have 8-1 =7

Therefore 7 is a factor of all the terms in (8 - 1)(8n-1 + 8n-2 + ... + 80).

Therefore 23n-1 is divisible by 7

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