8. Let n ≥ 2. Prove that (n+1)! + k is composite for k = 2,..., n + 1. This shows that there exist arbitrarily long intervals of composite numbers.

Q: Suppose that n pigeons are placed into m holes with m ≥ k(n+ 1) for some positive integer k. Suppose…

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Q: prove : 1+1/(3)2+1/(5)2+1/(7)2+...=(pie)2/8

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Q: Find a Big-O estimate of (nn+n2n+3n)(n!+3n)

A: Consider the function fn=nn+n2n+3nn!+3n The objective is to give big-O estimate for the given…

Q: An integer M is called B-power-smooth if every prime power pe dividing M satisfies pe ≤ B. For…

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Q: ) Prove every integer n >= 7, n! > 3n

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Q: Prove that the summation of three odd integers is odd.

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Q: A is the set of odd number greater than 5 and less than 23 n (A)=

A: A is the set of odd numbers greater than 5 and less than  23

Q: How many three-digit numbers are possible under each condition? (a) The leading digit cannot be zero…

A: We know that 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are digits in the number system.

Q: 13. Prove that 6 | (n³-n) for every integer n ≥ 0.

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Q: A sequence bo, b1, b2, ... is defined by letting bo = 5 and b = 4+bk-1 for each integer k> 1. Show…

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Q: Let F(n) denote the Fibonacci numbers. Prove the following property for n ≥ 3 directly from the…

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Q: Prove that n>= 12 is a sum of two composite number.|

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Q: of all positive integers Find the sum less than 200

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Q: Prove that for each odd natural number n with n greater than or equal to three,…

A: We have a product line called [(1 + frac12)]. (1 - \frac{1}{3}) (1 + \frac{1}{4}) [ ldots left(1 +…

Q: Show that the sequence delened by +5 mcreasing and ancg"for allkn an is thereas and anz9

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Q: Show that 10n <3n for n ≥no. Be sure to find no first.

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Q: How can the sum of k * (n choose k) = n(2 to the power of n-1) be proven akgebraically ?

A: Step 1: We have to prove that, Now by the binomial theorem, Now, differentiating the both sides of…