3. Answer any one of 3(a) and 3(b). (a) Use mathematical induction to prove the following inequality: 4n > 1+ 3n for all positive integers n. (b) Prove that E, i (2²-1) = (n – 1)2" + 1 for all positive integers n using mathematical induction.

Q: (a) Suppose that n > 1 is an integer with prime factorization n = pf"p.p. Prove that (k, + 1)(k, +…

A: As both of them are different questions(not a subpart of other), so as per the guideline, I am…

Q: Show that (2i – 1) (2n)! for every positive integer n!2" n. i=l

A: We have to solve given problem:

Q: Use mathematical induction to prove the following claim for all integers n > (2i - 3) = (n² – 2n +…

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Q: Prove that 1 + 2+22+23 + + 2" = 2n+1 1 for n 2 1. Use proof by induction.

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: Use induction to show that the following identity holds for all natural numbers: (n+1)(5n+6) Eo (3+…

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Q: 3. Use the Principle of Mathematical Induction to prove that for a natural number n, n(n + 1)(2n +…

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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: 1+ x Σ Aa" (1 – 2)? n=0

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Q: 1. Prove the following equality by induction: 1- 22 + 32 – 4? +. + (2n - 1)? - (2n)² = -n(2n + 1),…

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Q: 2. Using the mathematical induction, prove Σ (k² + 2k-2) = n(2n² + 9n - 5), k=1

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Q: This problem is much harder than it looks. Use induction to show that n³ <3n Vn ≥ 1 Hint: you may…

A: For   So it is true.

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: 1. Use mathematical induction or strong mathematical induction to prove the given statement. 1 (a).…

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Q: 4. Use mathematical induction to prove the truth of each of the following assertions for all n ≥ 1.…

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Q: 11. Prove that every positive integer n can be expressed in the form n=4k or n=4k+1 or n=4k+2 or…

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Q: Use the Principie to prove the Ving 3n+1 – 1 is divisible by 2 for all n > 0

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Q: For positive integers n and k (k sn+ 1), the kth term of (a + b)n is given by

A: The binomial expansion of (a+b)^n is

Q: (a) There exists two odd integers whose average is odd. 1 1 (b) There exists distinct positive…

A: (a) ans:- 1,5 (b) ans:- 2,3 &amp;6 For detail...go through the solution

Q: Prove the following fact by induction: n-1 For all n > 2, j. 4i 4+ 3n4" 4n+1 j=1 9.

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Q: Prove by induction that 3" <n! for all integers n with n ≥ 7.

A: As per the answering guidelines we will solve the first question only. Please upload the other…

Q: Let (M) be the statement P(n), n 2 1 where P(n) is 2[1+3+32 + ·.+ 3"] = 3n+1 – 1. (M) is a false…

A: For n=1, L.H.S. = 2[1]=2.

Q: (1) Prove by induction: Σ₁(i² − i + 1) = n³+2n

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

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Q: Arrange the steps in the correct order to prove that 21 divides 4n+1+52n-1 whenever n is a positive…

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Q: prove its true by mathematical induction

A: The mathematical induction technique is used to prove the validity of the statement for every…

Q: Prove by induction that for positive integers n, 13 | (4²n+1 + 3n−1).

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Q: (1) Prove by induction: Σ²₁ (i² − i + 1) = n³+2n

A: (1)  Let the given statement is  P n :  ∑i=1n i2 - i + 1 = n3 + 2n3             For  n = 1  we get…

Q: For each integer n2 2, let P(n) be the following equation (n – 1)n(2n + 5) (i – 1)(i + 1) = 72 - 6…

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Q: (Vn)(n2 +n+ 17 is prime)

A: Since, you have posted multiple questions , I am doing the very question (ques 7)  according to our…

Q: (2) Give a proof by induction of the formula: P(n): 12 +22 + ...+ n² = "(n+ 1)(2n + 1) 6

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Q: Select all of the following that are equivalent to >(2) +> (3k + 1)? k=0 k=0 п+1 A. O) (3k + 3) k=0…

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Q: -1/2 p. Find the coefficient of x" in the expansion of (1– 4x)2.

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Q: prove using induction: II ( 1 1 1 П 2k +1 2k+2, for n 20 (2n + 2)! k-0

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Q: Use proof by induction to prove that: =-n°(n+ 1)? %3D r=1

A: Mathemetical Induction

Q: Directions: Prove using induction. 3(5"+1 -1) 1. 3+3 5+3-52 +...+3 5" = for n a nonnegative integer.…

A: Mathematical induction

Q: 2. Prove by induction that 3² i=0 3n+1-1 2 for all n with n ≥ 0.

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Q: Use mathematical induction I to prove that for all integers n2 3, 2·3 + 3 · 4 + ·· ·+ (n – 1) · n =…

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