1. Prove that if 7n2 + 5 is even, then n is odd

Q: Prove the the following using either direct proof, indirect proof, or proof by contradiction. 2. n…

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

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Q: 7. Suppose that m, n E N are two consecutive numbers such that both m > n > 2. Prove that if n is a…

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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: Suppose n E Z. Prove that if n is even, then n2 + 5n + 4 is even.

A: Note: We know that the square of an even number is always an even number, since  Let x is an even…

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: , 7/4 /213, 33 b) Solve e In(9) + 21 = 41 for æ. a) Write In(- -) in the form a lIn(7) + bln(3)…

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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: 1. If today is Saturday, what day of the week will it be 1,500 days from today? 2. Prove that 4/2 +…

A: The explanation of the solution as follows:

Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

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: Prove that n²– 3n + 1 is odd for every integer n.

A: To prove that n^2-3n+1 is odd for every integer n.

Q: Prove by using proof by contradiction: If (3n)^2 is even, then n is even.

A: Any odd integer multiply by an odd integer is odd. Any even integer multiply by an even integer is…

Q: Show that if n is an integer and 3n3 +n is odd then n is odd, by using a) Proof by Contrapositive b)…

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Q: Disprove the claim: 3 m E N such that m > 3 and m? – 1 is a prime number.

A: The claim is given : ∃ m∈N such that m≥3 and m2-1 is a prime number Now, m2-1=(m-1)(m+1)--(1) Case…

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: 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 if n is an integer and 5n + 6 is even, then n is even.

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Q: (a) Prove that if n is an even integer then (n – 1)² + n is odd. (b) Given that n is an integer,…

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Q: Prove that 3" > n2 for every positive integer n.

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Q: The number of primes of the form n² – 6n + 5 where n is an integer is O 2 O 3 O 1

A: We have n2 -6n+5 ⇒n2-5n-n+5⇒n(n-5)-(n-5)⇒(n-1)(n-5)

Q: 4. Prove: if an/bn (Here, as always, L represents a finite number, not oo.) L, ba #0 for any n, and…

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Q: п 6. Prove that E(8i – 5) = 4n² -n for every positive integer n. п i=1

A: # Given: sum(8i-5) =4n^2-n  i=1,2,3,.....n  We have to prove the above equation??

Q: Show that if n is an integer and 5n + 6 is odd, then n is odd. (Use contradiction.)

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Q: (d) Prove that 2" +3" is a multiple of 5 for all odd n E N.

A: Use the mathematical induction to prove the given statement.

Q: Prove that if n is a positive integer, then n is even if and only if 7n + 4 is even.

A: (if part) : Let, n be an even positive integer. Then, n = 2k for some positive integer k. So, 7n+4 =…

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

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Q: 1. Let n be an integer. Prove that if n is even, then n2 + 2n + 3 is odd.

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Q: Prove that 3|(52n – 1) for all ne N.

A: We prove it by mathematical induction. 1st step We show that the statement is true for n = 1.…

Q: Suppose n is an integer. Prove that if n is even, then n²+3n+5 is odd.

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Q: 5 М — 11 Find formulas for the entries of M", wheren is a positive integer. M" =

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Q: Prove that 18 divides 2. 192n+1 16 . 174n+1 for all n e N.

A: We need to show that 18 divides 2·192n+1-16·174n+1 Now, a divides b is there exists an integer t…

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

A: If n is an an odd integer then we have to show that n4 + 4n2 + 11 is of the form 16k for some…

Q: Prove that the statements here are true for every positive integer n. 12 + 22 + 32 + · · · + n2=n(n…

A: Proved that for n≥1 fn=12+22+32+⋯⋯+n2=nn+12n+16 For n=1 LHS=12=1 RHS=nn+12n+16=11+12+16=66=1…

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

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Q: 7. Suppose a1, a2,...,an are integers, r is a solution to the equation x" + a1x"- + a2x"- +.. + an =…

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Q: 2. Prove that if 2n² + 7n is even and n is an integer, then n is even.

A: We have to prove that the given statement is true by the help of mathematical induction

Q: Let n e Z. Prove that if n is odd, then 5n + 13 is even.

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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 that n2 − 3n + 1 is odd for every integer n.

A: Prove the result using mathematical induction. P(n):n2-3n+1. Check whether the result is true for n…

Q: Prove by contradiction that 6n + 5 is odd for all integers n.

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

Q: (4a) Let n E Z. Prove that if n is even, then -5n – 3 is odd.

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Q: If n is an odd integer, show that n4 + 4n2 + 11 is of the form 16k.

A: Solution: The objective is to show that given equation is of the form 16k

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: 3. Let d = (a, n) where n > 1. Prove that if there is a solution to ax = b (mod n), then d divides…

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Q: In an inductive proof that for every positive integer n, n п(п + 1)(2n + 1) Σ 6. j=1 what must be…

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Q: There is an integer n such that 7n2 – 5n + 11 is a prime number. -

A: Let fn=7n2-5n+11for n=1f1=7×1-5×1+11      =13 , which is prime number for n=2f2=722-5×2+11…