1. Prove the following statements below. Consider n to be an integer. (a) If 3n + 2 is an odd integer then 9n +5 is an even integer. (b) If 9n +5 is an even integer then 3n +2 is an odd integer.

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

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Q: Prove the following statement. 1) if k is a positive integer, then 3k+2 and k+1 are relatively…

A: We have to prove: If k is a positive integer, then 3k+2 and k+1 are relatively prime.

Q: If n is an even postive integer, then 3n+2^n-2 is odd.

A: Number problem....

Q: 1. Let n be an integer. Prove by contradiction is even.

A: Given that n2+5 is odd. If possible let n is odd. Then we get, n = 2k+1

Q: 1. If n is an integer, then 2n +1 is odd.

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Q: Given n < 10" for a fixed positive integer n°2, prove that (n + 1)* < 10*1.

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Q: Prove that if n is an odd integer then 7n+5 is even.

A: Topic:- algebra  Note :- if more than one question uploaded than only first question need to be…

Q: 8. Prove the following statements: (a) If n is an even integer, then n2 - 6 is even. (b) If m is an…

A: We have to prove the given statements.

Q: V. Show for all integers n2 1 that 71(3*n+ – 52n-). а.

A: As you have asked more than one question from different backgrounds, I am solving the first question…

Q: If is an odd integer, show that n* + 4n2 + 11 is of the form 16k.

A: Firstly, we have to prove that n4+4n2+11 is in the of 16k for an odd integer n. for making the…

Q: If n is an integer, then 2n + 1 is odd

A: Given , n is an integer We have to show that 2n+ 1 is odd

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

A: To prove: 1+4+7+...+3n-2=n3n-12

Q: ! Prove the following statements. (a) If k is a positive integer, then 3k+2 and k+1 are relatively…

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Q: Prove that for any nonnegative integer n. n + +.. + 22m i+1 i +2 n+1 i=0 the

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Q: Prove that the statements here are true for every positive integer n. 4 + 42 + 43 + · · · +…

A: To prove: For every positive integer, 43+44+......+4n=4(4n-16)3

Q: 4. Prove that if n is an integer and n is a multiple of three (that is, n = 3k for some integer k),…

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Q: 2. Prove by contraposition: For all integers n, if (3n +4) is even, then n is even.

A: Contrapositive of 'if p then q' is if not q the not p'

Q: Prove the following statement by contradiction: If (n – 3)2 is an odd integer, then n is an even…

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Q: 1 Prove that -n³ + 1 n is an integer for any n e N. + 3 2

A: 13n3+12n2+16n=2n3+3n2+n6=n(2n2+3n+1)6=n(2n2+2n+n+1)6=n(n+1)(2n+1)6 To prove 13n3+12n2+16n is an…

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: Show that if the statement 1 + 2 + 22 + · · · + 2n - 1 = 2n is assumed to be true for n = k, then it…

A: Given, 1 + 2 + 22 + · · · + 2n - 1 = 2n ------------(1) is assumed to be true for n = k To prove -…

Q: Prove that if  n  is a positive integer, then  5 | (6^n − 1)

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Q: 1. Prove the following statements below. Consider n to be an integer. (a) If 3n + 2 is an odd…

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Q: Prove that the statements here are true for every positive integer n. 4 + 42 + 43 + · · · +…

A: To side equations are given, 4 + 42 + 43 + · · · + 4n=4(4n-1)/3We need to find out if above equation…

Q: Prove that the statements here are true for every positive integer n.   1 · 2 + 2 · 3 + 3 · 4 + · ·…

A: To prove: For every positive integer n, 1.2+2.3+3.4+.......+n(n+1)=n(n+1)(n+2)3

Q: 6.20. For which integers n > 2 does there exist a o e A, such that |o|> n?

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Q: Prove that for all integers m and n, if m is odd and the remainder is 2 when n is divided by 3, then…

A: Given, m is odd  When n is divided by 3, then the remainder is 2  We have to prove when  3m+2n  is…

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.…

Q: 7. For every integer n z 1, n(5n – 3) 1+6+11+16++(5n– 4) 2

A: Solution is in step 2

Q: R.4. Prove that for every integer n, n³ is even if and only if n is even.

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Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: 1. Prove the following using direct proof. a. If 5x – 3 = x + 5, then x = 2. b. If x and y are both…

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Q: 1. Prove: If n is odd then 5n + 4 is odd.

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

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Q: Prove the statement by: a) contraposition b) contradiction

A: Given statement is, “For all integers m and n, if mn is even then m is even or n is even.

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

A: Prove that the statement is true for every positive integer n. 22+42+62+...+2n2=2n2n+1n+13 We prove…

Q: * 10. Is the following proposition true or false? Justify your conclusion. For each integer n, n is…

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Q: Prove the following: for all integers n > 5, n2 < 2"

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Q: Show that for any integer n ≥ 2, C(2n,n) > C(2n,n-1)

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Q: Prove that the statements here are true for every positive integer n. 2 + 22 + 23 +……++ 2n = 2(2n -…

A: The given statement is: 2+22+23+...+2n=2(2n-1) We need to prove that the given statement is true for…

Q: 18. For every positive integer n, 2n < 2".

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Q: Prove that for every non-negative integer n the value of n^2+n+41 is prime

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Q: Show that if n is an integer, thenn= [n/2] + [n/2].

A: Floor function x: x gives as output the greatest integer less than or equal to x. Ceiling function…

Q: Prove that n^2 is an even number if and only if 3n + 1 is an odd number by proving  If n^2 is even…

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Q: (2) (a) Prove by contrapositive : Va, b e R* : a <b → a < a +b < b. (b) Prove by contradiction : For…

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Q: Prove the following statements: (a) If n is a positive integer then n 2 + 1 is an odd positive…

A: (a) n is an even positive integer. So, n = 2m for some integer m  So, n² + 1 = (2m)² + 1 = 2(2m²) +…

Q: 3n Prove that for every integer n > 0, 9|(4" + 8).

A: To prove: For every integer n≥0, 9|43n+8. Proof by induction method.

Q: Prove that the statements here are true for every positive integer n. 13 + 33 + 53 + · · · + (2n -…

A: Prove that the statements here are true for every positive integer n. 13 + 33 + 53 + · · · + (2n -…

Q: Prove or disprove the following: n+1 For any odd integer n, H= ()) 2

A: Let  n=2k+1         -  (1) Where k is an integer. Then n is any odd integer. Now, LHS= n24        -…