14. Prove that for every positive integer n, (n- 1)2"+1 + 2. %3D

Q: 28. Prove that for every integer n ≥ 1, 1+3+5++ (2n-1) (2n + 1)(2n + 3) + ··· + (2n + (2n − 1))* 1 3

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Q: Let n denote an integer. Prove that 5|n and 7|n if and only if 35|n.

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Q: 10 Re(1 + i)¹⁰ Select one: O a. 0 O b. 1/2 O c. 3/4 O d. 5/6

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Q: 16 M = 7 -5 Find formulas for the entries of M", where n is a positive integer. M" =

A: We have to find the entries of matrix Mn.

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: How many positive integers less than 10000 are there which contain at least one 7 or at least one 9…

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Q: Find the smallest positive integer n such that 5 (3² +2²) (3ª +2¹) (3³ +2³)... (3² +22") > 9256

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Q: In (5 – x) – In (5 + x) = -In 9

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Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

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Q: Prove that the product of any k consecutive positive integers is an integer multiple of k!.

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Q: Show that for positive integer n: (2n)! VT. 22n . n! Г 2

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Q: Show that for positive integer n,

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Q: n² + n+ 8 2 He has just shown you a convincing proof that if the equation is true for any integer k…

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Q: 32. Prove that 3 divides n³ +2n whenever n is a positive integer.

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Q: Prove that 1+2+3+4+...+n= n² + n 2 for every positive integer

A: We have to prove the statement using the principle of mathematical induction.We know thatPrincipal…

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

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Q: Find all integers 100 ≤ n ≤ 999 such that the last three digits of n² is same as n.

A: To find: All integers  such that the last three digits of is same as n.

Q: 1. Show there exists an integer n such that n² + 3n/2 = 1. Is the integer unique? Explain your…

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Q: Find a positive integer n such that 2n is a square and 3n is a cube.

A: Sol:- Let 2n be equal to x², where x is a positive integer. Then, n can be written as n = x²/2.…

Q: 5. Show that 2 32 3" 31 3" for all positive integers n.

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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: () is even. 3. Prove that for all positive integers n, the number

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Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n>1. All prime divisors of n!+1 must be bigger than n. This is…

Q: ) 6. Express the sum 9 + 28 + 65 + ... + (n'+1) using E-notation.

A: We have to find sum of series:

Q: FIND AUB,GIVEN THAT A= {x/x is a positive integer. 2<x<6}    B= {x/x is a negative integer, - 6<x<…

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Q: Prove that the statement is true for every positive integer n.

A: We know that the sum of a geometric series of the form: a+ar+ar2+ar3+...........+arn-1 with the…

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

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Q: (d) (11 × 14" + 1) is a composite integer for every positive integer

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Q: 6: Prove by contradiction that for any positive integer x, x+ 49 x ≥ 14.

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Q: Suppose that 75% of students attend lectures for a particular course and 20% of students fall the…

A: Conditional probability : Conditional probability is a measure  of the probability of an event…

Q: Prove that 3 divides n³ +2n whenever n is a positive integer.

A: Introduction: Consider a starting point for which the assertion is correct. The proposition is true…

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

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Q: 5) Let n be a positive integer such that 7 | (8-1). Prove that 7 | (8=*1 - 1). (Hint: Rewrite 8*1…

A: Given: Let n be a positive integer such that 78n-1. To prove: 78n+1-1

Q: 4. Find all the prime numbers that can be expressed in the form n2 – 9 for some integer n.

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Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: Prove that 6">5" + 9 for any integer n ≥ 2.

A: Mathematical induction is a technique used to prove statements about well-ordered sets of natural…

Q: 5. Prove that the integer 3". 22" – 1 is divisible by 11 for all n > 0.

A: