Prove the following formula for all integers n 21: 1/2+2 3+1 4+... 2.3 + n.(n+1) || n n+1

Q: For a natural number n, we define n! = 1·2·3· ... · (n − 1) · n. For example: 2! = 1 2,3! = 1 2 3 =…

A:

Q: 1. Prove that nº + n is even for every integer n.

A:

Q: Claim: For every odd integer n, n² - 1 is a multiple of 8. Purported proof: Suppose there is an…

A: The given claim is "For every odd integer n, n2−1 is a multiple of 8".Purported proof:Suppose there…

Q: Prove for all integers n ≥ 1 that 1(2) +2(3) +3(4) + ... + n(n + 1) = n(n+1)(n+2) 3

A:

Q: Prove 0 + 1 + 2 + 3 + ...+ n = (n(n + 1))/2

A:

Q: Prove the following statement: for all integers n> 1 1³ +2³+ 2 = [ n(n+1)]² 2 + n³ = [ 3

A: To Prove: The following statement: for all integers n≥1 13+23+...+n3=nn+122

Q: For any integer n ≥ 1, C(6n, 3n) C(6n, 3n-1). O True O False

A:

Q: Prove that 2n+1 +(-1)" 3-2 whenever n is a nonnegative integer.

A:

Q: For every positive integer n, 1 + 1.4 4.7 + ... + 1 (3n - 2)(3n+1) n = 3n+1

A:

Q: For every positive integer n     12+22+32+...+n2=n(n+1)(2n+1)/6 Let S(n)=12+22+32+...+n2 How do…

A: See the attachment

Q: Consider the two statements: P: For every integer n there exists an integer m such that m + n = 0.…

A:

Q: Prove the following statement by contradiction: If (n – 3)2 is an odd integer, then n is an even…

A:

Q: Prove that 8 | (54" +7) for every positive integer n.

A:

Q: Prove for any natural number n that 12 + 22 + 32 + ...+ n² = ÷n 1 (n + 1)(2n + 1).

A: Step 1: This can be proved by using Induction method. This method involves following steps. Prove…

Q: 4. For each integer n with n > 2, let P(n) be the formula n-1 n(n – 1)(n + 1) Eii + 1) = iml a.…

A:

Q: 3. Prove that 2n > n² for all integers n such that n > 4.

A:

Q: Find all positive integers n that is a product of two distinct primes such that o(n) = 60.

A:

Q: Prove that: 3n > 2n + n2 for all integers n is greater than or equal to 2

A:

Q: Show that there is no positive integer n for which In-1+ In+1 is rational.

A:

Q: Prove that 6 divides n3-n for all non-negative integers n.

A: See the detailed solution below.

Q: Find the smallest positive integer N≥ 2 that gives remainder 1 upon division by each of the integers…

A: