Prove by mathematical induction that for all positive integers n 1: a 12 x 2 + 22 × 3 + 32 × 4 + · ·. + n2 (n + 1) (n + 1) (n + 2) (3n + 1) b 1 x 22 + 2 × 3² + 3 × 42 + ... + n(n + 1)2 = bn(n + 1) (n + 2) (3n + 5)

Q: If n2 4 is an integer,prove by induction that n<nl.

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Q: k+4 n(3n +7) k(k +1)(k + 2) 2(n+1)(n+2) k=1

A: Given ∑k=1nk+4k(k+1)(k+2) To show  ∑k=1nk+4k(k+1)(k+2)=n(3n+7)2(n+1)(n+2)

Q: (a) Prove that (-1+iv3)" + (-1– iv3)" has either the value 2"+1 or the value –2" if n is any integer…

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Q: Prove that 5n-1 is divisible by 4 to any positive integer n

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

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Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

A: The objective is to prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

Q: (a) Use mathematical induction to prove that 5n+1 + 2(3n) + 1 is divisible by 4 whenever n is a…

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Q: Use induction to prove that 3 | (10n+1 + 10n + 1) for all positive integers n

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Q: divisible by 3 for

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Q: Use induction to prove that 3 | (2²n — 1) for all nonnegative integers n.

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Q: Use cases to prove that for all integer n, n2+3n+5 is odd.

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Q: Can you assist  me in  verifing    that    for    all    n ≥ 1,    the    sum    of    the    first…

A: Here, we have to verify that ∀n≥1, the sum of the squares of the first 2n positive integers is given…

Q: ve or disprove: For all positive integers n, if 771 n, then 7(77n) = 47(n)

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Q: 17. Prove by induction that 1³+23++n³= (1+2+...+n)² for all positive integers n, that is, the sum of…

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

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Q: For every integer n > 1, 2+4+6+ + 2n = n² + n.

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Q: Prove by induction that (2n+1)3 is an odd number for all positive integers n.

A: First principle mathematically induction : Suppose is a mathematical statement .Here n is a natural…

Q: 1. Prove by mathematical induction that n(n+1) (a) 1+ 2+ + 12 = 2 (b) 1+3+5++ (2n-1) = n²…

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Q: Prove or disprove: For all positive integers n, if 77n, then 7(77n) = 47(n).

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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 that 4 divides 6 7" - 2.3" for all integers n 20.

A: We need to prove that 4 divides 6.7n-2.3n for all integers n≥0.

Q: Prove that n3+(n+1)3+(n+2)3 is divisible by 9 for all non-negative integers  n.

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Q: Prove that 4 does not divide n2 +2 for any integer n.

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Q: Use proof by cases to prove that for any integer n, n² + 3n + 7 is even.

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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: (g) Prove by induction that for any positive integer n, n(n + 1)(2n + 1) Wi

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Q: (c) Prove that n? + 1 is not divisible by 3 for every integer п.

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Q: Use the method of proof by cases to prove that for any integer n the product n(n + 1) is even

A: Given,  for any integer n the product n(n + 1) is even  We have to prove it using the method of…

Q: C) Prove that the product of k consecutive positive integers is divisible by k!

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