Prove by mathematical induction that for all positive integer values of n: n a 2 (r3 – r) = }(n – 1) (n) (n + 1) (n + 2) - r=1

Q: Prove n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

A: We have to proff the question

Q: Establish the formulas below by mathematical induction: 1^3 + 2^3 + 3^3 + .... + n^3 = [ (n(n+1))/2…

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Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: 3. Using induction, verify that 12 + 3 + 5² + (2n – 1)² = n(2n–1)(2n+1) is true for every positive…

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

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Q: (Prove by induction that for all integers n ≥ 1, 1² + 32 +5² +...+(2n-1)² = n(2n-1)(2n + 1) 3

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

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Q: 21 divides 4(n+1) + 5(2n-1) whenever n is a positive integer.

A: Steps and explanations are as follows:In case of any doubt, please let me know. Thank you.

Q: Let $n$ be a positive integer. Among C(2n,0),C(2n,1),..., C(2n,2n), C(2n,n) is the largest. True…

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Q: Prove that 25|(6" – 5n – 1) for all positive integers n. -

A: We can prove given statement by principle of mathematical induction: Alogrithm: Obtain P(n) and…

Q: 2n+1+(-1)" Prove that for all integers n 2 0, 3.2"

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Q: 1.-Proof by mathematical induction that 5 + 9 + 13 + ··· + (4n + 1) = n(2n + 3) ...

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Q: b) Prove that 4" 10 by induction

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Q: Use cases to prove that for all integer n, n2+3n+5 is odd.

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

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Q: Prove (using induction) that for all integers n ≥ 1 1 +6+11+ 16 + + (5n-4) · ... = n(5n - 3) 2

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Q: induction to prove that 12 + 22 + ... + n2= n(n+1)(2n+1)/6  for all n>0.

A: To prove by induction :Let the proposition P(n): 12 + 22 + ... + n2 = n(n+1)(2n+1)/6Here n is…

Q: 17. Prove by induction that 1³+23++n³= (1+2+...+n)² for all positive integers n, that is, the sum of…

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Q: 1) Let x be a positive real number. Show by induction on n that (1+x)" > 1+ nx for all integers n >…

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Q: Use induction to prove that 12 + 22 + ... + n2= n(n+1)(2n+1)/6  for all n>0.

A: As per the question we have to use induction to prove that :12 + 22 + ... + n2 = n(n+1)(2n+1)/6

Q: Use induction to prove that the sum of {n=-2} and {N} (4n − 5) = (n + 3)(2n − 9).

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Q: I. Induction a. 1 + 2 + 2² +: · +2" = 2n+1 _ 1, for all nonnegative integers n.

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Q: S Prove by mathematical induction that x 1 is a factor of x" 1 for all integers n > 1.

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Q: 1. Prove by mathematical induction that n(n+1) (a) 1+ 2+ + 12 = 2 (b) 1+3+5++ (2n-1) = n²…

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Q: (3n)! is an integer for all n > 0. (3!)n Prove that

A: We will prove it by Mathematical induction. First We show it is true for initial condition n=0…