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

Q: Prove that if n is an odd integer, then n3 +n+1 is also an odd integer.

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

A: Given, 3+9+15+..+(6n-3)=3n2

Q: Prove that the analogue of lemma 1.23 is not true for numbers of the form 4n+3 where n is an…

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Q: Prove or disprove For every integer m, m(m+1) is even

A: To prove or disprove the given statement- For every integer m , m(m+1) is even .

Q: Prove or disprove: For every positive integer n, 22n – 1 is divisible by 3. -

A: I have proved the result using mathematical induction

Q: Determine for which positive integer values of n, 3n3 +2<n* and prove your claim by mathematical…

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

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Q: 2n 10. Prove that 31 (5 - 1) for every integer n 2 0.

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Q: -ん 2 kodd

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

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

A: Given,  2+7+12+...+(5n-3)=12n(5n-1)

Q: Prove that n2 – 7n + 12 is nonnegative whenever n is an integer with n > 3.

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Q: 3) Prove by induction that n²>2n + 3 for every integer n >3

A: According to the guidelines, we can solve only one question. So, I solve question number 3. Please…

Q: Suppose thatk and n 2k are fixed positive integer. Justify the identity k+1 jk

A: to prove ∑j=knjk=n+1k+1above equation can be written as S2(j,k)=S2(j-1,k-1)+k.S2(j-1,k)count the…

Q: 7) Prove that for any integer n, n' -n is even.

A: We need to prove that for any integer n, n3-n is even, Now, an integer can be either even or odd.…

Q: . Prove that 3| (52n – 1) for every integer n 20.

A: Solution:  To show :   3|(52n-1)     , for every integer n≥0

Q: Let n be a positive integer. Prove that (:) n +2. +...+n. n2"-1.

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Q: 2. Prove by induction (n+1) (n+2) - 1 is odd for all positive numbers of n >= 0

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Q: b) Prove that () - 2) = n², where n is a positive integer.

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Q: Prove that there exists a positive integer n > 2 such that 4n+1 is a prime number.

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Q: Prove that n- + 1> 2" when n is a positive integer with 1 <n<4.

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

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

A: we have to prove that the statement is true for every positive integer n. 4 is a factor of 5n-1

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

A: See the detailed solution below.

Q: Prove that 3 | 22n+1 +1 for all natural numbers n.

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Prove that there is some positive integer n such that n, n + 1,n + 2,... , n + 200 are all…

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Q: prove that it ņ is an 3n +2 is odd then integer and is add

A: This is related to Number System

Q: Prove that if n is an integer and 3n + 2 is even, then n is even using a) a proof by contraposition

A: Given , n is an integer .To prove → if 3n+2 is even , then n is even using a proof by composition…

Q: Prove that for every positive integer n, there are n consecutive composite integers.

A: This question is related to Descrete Mathematics

Q: Use proof by cases to prove that for any integer n, n² + 3n + 7 is even.

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Q: 4) Determine if the following statement is true and either prove it or provide a counter example:…

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Q: Prove or disprove that for every positive integer n, if n is even then n^{2}+1 is prime.

A: False

Q: Suppose x is a fixed real number with x 2 -1. For n2 1, prove using mathematical induction: (1+ x)"…

A: Here we will show that given statement is true by mathematical induction method.

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

A: Given, Prove this statement is true for every positive integer n: 2+4+6+..+2n=n(n+1)

Q: Prove that 11 is the only prime number of the form 4n2 - 25.

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Q: Prove that 74n+3 + 2 is a multiple of 5 for all non-negative integers n.

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Q: Prove that n2 +1 > 2? when n is a positive integer with 1<n<4.

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Q: Prove that 2n + 3 ≤ 2n if n is an integer greater than 3.

A: Sol :- To prove:- 2n+3&lt;=2^n if n is an integer greater than 3 We prove this by induction For n=4…

Q: Show that if a is a positive integer, then Lo-:(---) n(n +a) n=1

A: Let a is a positive integer, the term a can be written as a = n + a – n

Q: prove that there is no positive integer n such that n^2+n# = 100

A: n^2+n^3 = 100 We can evaluate this till some positive integers

Q: Prove 8^ +l is composic for any poSitive integer n.

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Q: prove that (n(nt)* s an integer V impers

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Q: 2. Prove that a positive integer n is even if and only if nis even.

A: We prove that a positive integer n is even if and only if n2 is even.

Q: Prove by direct method: for any n integer,if n is odd then 3n+2 is odd

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Q: Prove by induction that (2) = n22"+1 – n2"+2 +3.27+1 - 6 whenever n is a positive integer

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Q: =) Prove that (*) + ("*) =n“, where n is a positive integer. n", where n is a positive integer. 2

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Q: Prove that if n is an odd integer, then 3n+ 5 is an even integer.

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