that for all integers n, it is the case that n is even if and only if 3n is even. That is, prove both implications: if n is even, then 3n is even, and if 3n is even, then n is even.

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

A:

Q: Prove the theorems: i) "If n is an integer, and if n² is even then n is even". ii) "If n is an…

A:

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

A: To prove i - if n is integer and 3n3+n is odd then n is odd a) By contrapositive :   (P→9)≡(~q)…

Q: Prove the following statement by contradiction. For any integer n, if n² + 4n + 5 is even, then n is…

A:

Q: Prove that if n is an integer and n3 + 5 is odd, then n is even

A:

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

A: Given: 3n + 2 is an even integer Let n is not even, thus n is odd. Therefore, by using the…

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: Prove that if n is a positive integer, then n-1 is odd if and only if 5n + 13 is odd.

A: Given that n is any positive integer. Now, to prove that n-1 is odd iff 5n+13 is odd

Q: Prove that by mathematically Induction 2n < (n + 1)!, for all integers n ≥ 2.

A: Let s(n):2n&lt;(n+1)!,n≥2...(1) Now, Substitute n=1 in (1), so, 2(2)≤(2+1)!⇒4≤3!⇒4&lt;6 Which is…

Q: Prove that if n is an integer, then 3n? + n + 14 is even

A: Case I - Let n be an odd integer , then n= 2k+1 where k is an integer. Then 3n2+n+14   =…

Q: Given positive integers k and m, there exists an integer N ≥ m such that N →(m)k

A:

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169. - -

A: The given expression is: 33n+3-26n-27 This is to be proved that the expression is divisible by 169.

Q: For all integers n, if n2 is odd then n is odd.

A:

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

A: Given arithmetic progression (AP) is: 2+5+8+.........+3n-1 Here the first term is 2 and the common…

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

A:

Q: Prove that for all natural numbers n, (n^3/3) + (n^5/5) + (7n/15) is an integer

A:

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

A:

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

A:

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: If you were going to use a proof by contradiction to prove the following theorem, which of these…

A:

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

A:

Q: Prove that if n is an integer and n² is an even integer, then n is an even integer

A:

Q: Prove that by mathematically Induction 2" < (n + 1)!, for all integers n2 2.

A:

Q: 8) For all integers n, if 3n + 2 is even, then n is even.

A: To show if 3n + 2 is even then n is even   "Since you have asked multiple question, we will solve…

Q: Prove that “if n3 + 5 is odd, then n is even” by using the following: a) Direct Proof b) Indirect…

A: If n3 +5 is odd then n is even. Given:  n3 +5 = 2K+1                (K is any integer)       n3  =…

Q: 4. Let n and k be two positive integers with n 2. Prove that (n 1)² divides n-1 if and only if n -1…

A: The given problem is to prove the given following statement. Given statement, let n and k be two…

Q: Prove that for all integers n, if n² is even then n is even.

A:

Q: Prove that for all integers n, n2 – n + 3 is odd.

A: Suppose n is any integer .By the quotient remender theorem with n=2 ,n is either even or odd. Case…

Q: Prove that if m and n are integers and mnis even, thenm is even or n is even.

A:

Q: 2. Prove that if n is even, then 3n + 1 is odd

A: The given expression is 3n + 1. Prove that 3n + 1 is odd, if n is even.

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

A: False

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

A:

Q: Prove that if the sum of S = a1 + a2 + … + an is an even integer, and all the terms a1, a2, …, are…

A: Given S=a1+a2+…+an is an even integer and all the terms a1, a2,…,an are odd integers. To prove that…

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

A: The given statement is,

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

A:

Q: Prove that n is even if and only if n2 is even.

A:

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…

A:

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

A: 13+23+33+---+n3=nn+122 Let us use the Principle of Mathematical Induction to prove the given…

Q: Give a contrapositive proof of the theorem  If n is an integer and 3n+2 is even ,then n is even

A: We have to prove given statement by contrapositive proof.

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 that for any integer n, if n³ is odd then n is odd.

A:

Q: Prove that by mathematically Induction. 2^n < (n + 1)!, for all integers n ≥ 2.

A: We will prove the base case for n=2 Hence 22&lt;(2+1)!4&lt;3!4&lt;6 True Now assume the result is…

Q: Prove each of the following statements. 1. 2 divides n 2  + n whenever n is a positive integer.

A: Solution 1: Given statement is 2 divides n2+n, n∈ℕ Let p(n)=n2+n For n=1,  p(1)=12+1=2 So 2 divides…

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

A:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.