Prove that n2+n  is  divisible by 2, where nN.

Q: Prove that 2n32n - 1 is always divisible by 17.

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Q: Let n be an integer. Prove that if 2n3 +4 is divisible by 4, then n is even.

A: n is an integer. To show: 2n3 + 4 is divisible by 4 then n is even.

Q: (a) Show that n(n2 – 1)(n + 2) is divisible by 4 for all integer n.

A: As per guidelines , if there are more than one questions , we have to solve the first question only…

Q: Show that for any integer n, n² – 2 is not divisible by 3.

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

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

A: To show  33n+3 - 26n - 27 is divisible by 169.

Q: Prove that vn if n is divisible by 3 then n2+2 is not divisible by 3

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Q: Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.

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Q: 1. Prove that 7" -1 is divisible by 6 for all n > 1.

A: As per Bartleby Guidelines we can solve only one question at a time. Hope you understand

Q: Prove that for all integers n.4n² +6n + 1 is not divisible by 4. Be sure to explicitly show how each…

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Q: Q2 Let n be an integer. Prove that if 2n3 +4 is divisible by 4, then n is even.

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Q: show that n(n^2-1)(n+2) is divisible by 4 for all integers n

A: To show that n(n^2-1)(n+2) is divisible by 4 for all integers n

Q: Prove that for any integer n, we have n - 2 is not divisible by 4.

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Q: Prove that (n – 1) +n° + (n + 1)° is divisible by 9 for n E {1,2,3, . }

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Q: Show for all integers n> 1 that 7 I (3ªn+l – 5²n-1 ).

A: a) Prove 734n+1-52n-1 for all integers n≥1 by mathematical induction as follows. Basis step: For…

Q: For any non negative integer n , prove that an − bn is divisible by a−b,where a and b are any…

A: answer

Q: Show that (n + 1)!)" < 2! · 4! - (2n)!.

A: The objective is to prove the given statement using induction

Q: If n is a positive integer prove that 3^(2n) - 26n -1 is divisible by 676

A: if n is a postive integer prove that 3^(2n) - 26n -1 is divisible  by 676

Q: Prove that for any n E N, nº + 2n is divisible by 3.

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Q: Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the remainder…

A: we have to Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the…

Q: Use the Principie to prove the Ving 3n+1 – 1 is divisible by 2 for all n > 0

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Q: Prove that for all positive integers n, 4n-1 is divisible by 3.

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Q: Show that 3·4n+ 51 is divisible by 3 and 9 for all positive integersn

A: Consider the provided expression,

Q: For all integers n> 0, 32n – 1 is divisible by 8.

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Q: Prove by mathematical induction that for all integers n 2 0: a 5" + 2 x 11" is divisible by 3, b 33n…

A: (a) Case 1 : n=1 51+2×111 = 5+2×11= 5+22= 27 As 27 is a multiple of 3 , hence the result holds for…

Q: Prove that there is no integer n for such that 5n+3 is divisible by 5.

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Q: Prove that if q^2 is divisible by 3, then q is divisible by 3.

A: To prove that if q2 is divisible by 3, then q is divisible by 3.

Q: 3. Show that 13" – 7ª is divisible by 6 for n > 1.

A: As per Bartleby guidelines we can solve any one question at a time. Hope you understand.

Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: Prove that2.7n + 3.5n – 5 is divisible by 24, for all n ∈ N.

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Q: If n = 4k + 3, does 8 divide n2 - 1? (Assume that n and k represent integers.) O Yes, because n2 - 1…

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Q: Let x € Z. Prove that if x² - 8x + 11 is divisible by 4, the x6- 1 is also divisible by 4.

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Q: Let x Є Z. Prove that if x2 - 8x + 11 is divisible by 4, the x6-1 is also divisible by 4.

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Q: Prove that Vn if n is divisible by 3 then n2+2 is not divisible by 3

A: If a and b are two integers. Suppose n is non zero integer such that |n|≠1. If a is divisible by n…

Q: Show that 3-4" + 51 is divisible by 3 and 9 for all positive integers

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Q: Find all integers n> 1 such that 1" + 2"+... +(n – 1)" is divisible by n.

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Q: The binomial theorem states that for any real numbers a and b, (a +b)" = E o (") a"-* for any…

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Q: Prove that for all n E N, 7" +2 is divisible by 3.

A: We will use mathematical induction to prove the result in question 4.

Q: Show that 3-4" + 51 is divisible by 3 and 9 for all positive integers n.

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Q: Find all positive integers n for which n1º +1 is divisible by 10.

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Q: 1. Use Direct Proof to prove that the product of two consecutive integers is divisible by 2.

A: We have to prove that product of two consecutive integers is divisible by 2.

Q: Prove that for any odd positive integer, n, the sum of n consecutive positive integers is divisible…

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Q: How many terms will be in the expansion of (2x+3y)^6?

A: How many terms will be in the expansion of (2x+3y)^6?   Use the binomial expansion theorem to find…

Q: 2. 6n – 1 is divisible by 5 for n any positive integer

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Q: Obtain the expansion of : 2 up to the term in x8 1-3x2

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Q: Use the {\bf Cayley Hamilton Theorem} to come up with an expression for A" in terms of n when 1 A =

A: As you have asked more than one question, we shall answer first Que only. For others kindly post…

Q: Is it true that for any odd n the number 2" +1 is divisible by 3? Prove your answer.

A: We prove this by using PMI(Principle of Mathematical induction)

Q: Prove that n(n + 1)(n + 2) is always divisible by 6

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Q: For any non negative integer n , prove that an – bn is divisible by a-b,where a and b are any…

A: Use the binomial factorisation

Q: divisible

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