For n ≥ 2 , Prove  that  (2 choose 2 ) + (3 choose 2 ) + (4 choose 2 ) + ....... + ( n choose 2 ) = ( n+1 choose 3 ) [Hint: Use induction and Pascal's rule.]

Q: 10. Using the first principle of math induction, prove that, for n positive integer n³ + 2n is…

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

A: We have to proff the question

Q: 3. Suppose that a store offers gift certificates in denominations of 25 dollars and 40 dollars.…

A: We need to determine which amounts can be made from 25 dollars and 40 dollars. Since 25 and 40 are…

Q: 3. Prove or disprove. For n> 1, use mathematical induction to establish each of the following…

A: Let's say we need to prove a formula or a statement P(n). To prove this, we need to prove that The…

Q: his is from a pretty basic proofs class. Prove True or False: if a|(bc), then a|b and a|c . I've…

A: False

Q: Use mathematical induction to prove that 2+4+6+...+ 2n = n(n + 1)

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Q: Prove by induction that (1)1! + (2)2! + (3)3! + ... + (n)n! = (n where n! is the product of the…

A: A mathematical proposition is given.Let P(n):1.1!+2.2!+3.3!+...+n.n!=(n+1)!-1.Here, P(1)=1.1!=1 and…

Q: Part A Prove the following theorem using either regular or strong induction. Theorem: Suppose that n…

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Q: Prove (m) = m! n! (m-n)! using mathematical induction.

A: It shows above that the formula holds for k+1. By the principle of mathematical induction, the…

Q: "For all integers a, b and c, if a | b and b | c, then a | (5b – 3c)"

A: 4. Given the integers a, b and c. To prove: If a/b and b/c, then a/5b-3c.

Q: Induction: m (2n-1); nz n=? • Prove by Mathematical 1+5+9+...+(4n-3)

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Q: 17. 1+3n 0.

A: Alogrithm: Obtained P(n) and understand it's meaning. Prove that statement P(1) is true. Assume…

Q: Q1. Let x and y be two integers. Prove that for all integers m 2 1, x™ – y™ is a multiple of x-y…

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Q: Use induction to prove the following for all natural numbers n: 1²+2²+3²+ n(n+1) (2n+1) +n n² = 6…

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Q: . Use mathematical induction to show that n Σ r=1 (2r − 1)² = 1² + 3² +5² + + (2n − 1)² = = n(2n −…

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Q: 5.2.8 More generally, use induction to prove that (a - b) | (a" – b") for any positive integers…

A: Given: (a-b) | (an-bn) ----(1) Substitute n=1 in equation (1): (a-b) | (a1-b1)=1 The assertion is…

Q: Define 0! = 1. For any positive integer n, define n! = n- (n- 1)!. For example, 6! = 6x5x4x3x2×1=…

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Q: Show that 1+5+9+...+ (4n-3) = 2n²n using mathematical. induction.

A: Base case:    For n = 1:          LHS = 1          RHS = 2.12 - 1 = 2 - 1 = 1      So, LHS = RHS.…