Prove using PMI: For all positive integer n, 1+4+7+ ...+(3n-2) = [n (3n-1)]/2

Q: 3n > n for every positive integer n.

A: Now, since we assumed 3k>k, we can substitute k with 3k in the inequality: 3k+1>3k Since ,…

Q: 1.) Prove every integer n >= 1, 1+5+9...+(4n-3) = n(2n-1).

A: 1.Apply mathematical induction to prove the statement. Let the given statement be Pn:…

Q: prove that n(13n²-1) is divisible by 6 for every n ≥1

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Q: Prove an+1 = (an2+ 1) / 2 with a1 = 2 is monotonic increasing.

A: Solution:-

Q: 11. Prove that every positive integer n can be expressed in the form n=4k or n=4k+1 or n=4k+2 or…

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Q: given 7(x-1)=2(3x+2) prove x=11

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Q: Prove that 2n + 3n is a multiple of 5 for all odd n € N.

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Q: Prove for any natural number n that 12 + 22 + 32 + ...+ n² = ÷n 1 (n + 1)(2n + 1).

A: Step 1: This can be proved by using Induction method. This method involves following steps. Prove…

Q: Suppose that f (n) = f (n∕5) + 3n2 when n is a positive integerdivisible by 5, and f (1) = 4. Finda)…

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Q: Show that any positive integer n with n ≥ 4 can be expressed as n = 2x + 5y where x ≥ 0, y ≥ 0.

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Q: A positive integer n is called superperfect if ơ(ơ(n) = 2n Show that 16 is superperfect.

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Q: Euler's formula for generating amicable pairs of integers is the following rule: = Suppose n> m > 1…

A: Note : In the Euler's formula for generating amicable pairs it is r = 2n+m (2n-m + 1)2 - 1 and not…

Q: 7. For n ≥ 1, verify that 12+32 +52 ++ (2n- +3²+5² · + 2n-1)² = (2n+1) 3

A: Ans : page 1:Ans page 2:

Q: ndicate the least integer n for which (3x5 + x2)4 is O(xn).

A: Solved using the binomial theorem

Q: Deduce from Q2 that if an integer N can be written as N = x² + ny² , and if a prime number q | N can…

A: We will use standard division algorithms to deduce the given statement.

Q: For each integer n2 2, let P(n) be the following equation (n – 1)n(2n + 5) (i – 1)(i + 1) = 72 - 6…

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Q: I. Induction a. 1 + 2 + 2² +: · +2" = 2n+1 _ 1, for all nonnegative integers n.

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Q: 12 + 22,., +n? =n(n+ 1)(2n + 1), Vn eN

A: We will prove this as following.