Use mathematical induction to prove that, for n≥ 1, n³-n is divisible by 3. Note: If you would rather show your work on a separate sheet of paper, you may up- load your work in Question 12.

Q: 1. Show that the principle of mathematical Induction implies that I is the Ieast positive natural…

A: To find gcd(493,899)

Q: Suppose that A, B, C, and D are digits from 1 to 9 and that A + B = CD (a) What digit must C be?…

A: Given that  A, B, C and D are digits from 1 to 9 such that  A + B = CD   (a)  If 1 ≤ A, B ≤ 4 then A…

Q: 1 Use Mathematical Induction to prove that 2 k2 k=1

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Q: C. For any natural number n, let Pn be the statement that n³ + n + 1 is even. i. Show that for every…

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Q: Question 3*. Prove, using mathematical induction, that 8" n 2 1. 3"is divisible by 5 for all…

A: 3.

Q: 1. (Divisibility) Let a, b e Z with a # 0. We say that a divides b if there is some integer k such…

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Q: 2k > 1+ k

A: We have to prove 2k ≥ 1 + k for every natural number k

Q: Use the principle of mathematical induction to prove the following. For any ne N, 3 divides n n. For…

A: We will prove the given statement using mathematical induction method.

Q: The remaining problems require you to construct a mathematical proof by induction. Remember, if you…

A: Mathematical induction is used

Q: Prove by induction that it is possible to pay, without requiring change, any whole number of…

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Q: Prove that for any integers,s(s−1)(2s+ 5) is divisible by 6

A: Given that, s(s−1)(2s+ 5) we know that product of three consecutive integers is divisible by 6.…

Q: Suppose each license plate in a certain state has three digits followed by three letters. The digits…

A: The objective of this question is to find out the total number of license plates that can be…

Q: 1. Establish by mathematical induction that any postage of at least 12¢ can be obtained using 3¢ and…

A: For the solution follow the next steps.

Q: 2. Use the Principle of Mathematical Induction to prove for any natural number n that: 2 (1– )(1 n2

A: In order to prove a relation p(n) for all natural number n we follow some steps: 1. We check whether…

Q: Question 2. Prove by induction that 52+1 +22+1 is divisible by 7 for all n E N.

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Q: 4. Use mathematical induction to show that for any integer n ≥ 0,8" - 2" is divisible by 6.

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Q: Use the Principle of Mathematical Induction (PMI) to prove the following for all natural numbers n 1…

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Q: Is there any derivation of the formula of permutation and combination?

A: Derivation for the formula of permutation and combination . Permutation : (Arrangement of r things…

Q: Problem 8. Use the Principle of Mathematical Induction to prove that 2" < n! if n is an integer…

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Q: 10n+1 + 10" +1

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Q: Prove by induction that for any regative integer não and a ³00 1 non

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Q: 3. Let n E 00. Prove by induction that n² - 1 is divisible by 4. (Hint: You cannot prove this by…

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Q: range the given steps in the correct order to prove that Rank the options below. k + = 1-2 k-(k+1)…

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Q: 2. Use induction to prove that 9" – 1 is divisible by 8, if n E N!

A: We need to use induction to prove 9n-1 is divisible by 8 . Now , we use induction hypothesis to…

Q: Give a direct proof of the claim: V n E N, Vk E N, if the last k digits ofn are divisible by 2k then…

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Q: 5. Use mathematical induction to show that for any integer n ≥ 0,7" - 1 is divisible by 6.

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Q: Prove the following fact by induction: For all n > 7, n! > 3".

A: We Know that The Principle of Mathematical Induction(ii) If the statement is true for n = k (where k…

Q: use your own words to describe the steps of the method of mathematical induction

A: We have to write the steps for method of mathematical induction

Q: A. The number of permutations of length n that avoid 132 and have the additional property that their…

A: Based on the additional property, the first entry is larger than the second entry is Cn-1, (the…

Q: 19. Prove 1+ 10 + 102 + 103 + 10* + .+ 10k-1 = (10k – 1) using the principle of ... mathematical…

A: we have to prove by mathematical induction