The factorial of a positive integer n, denoted n!, is equal to the product of all integers between 1 and n. That is, n! = n (n- 1) (n - 2) .. (2) · (1). For example, 1! = 1, 2! = 2 ·1 = 2, 3! = 3- 2 1= 6, 4! = 4 . 3· 2. 1 = 24, etc. As a result, one can show that n! 2 n for all n 2 4 (try calculating n! and n for a few values of n 2 4 to convince 1 converges or diverges. n! n=1 yourself of this). Use this and the Comparison Test to determine whether O a. The series converges. O b. The series diverges. iM:

Q: Given that the expression (n²+n+1) n! can be written as a(n + 2)! + b(n + 1)! + cn! where a, b, c…

A: The main objective is to find the sum value.

Q: How many positive integers less than 10000 are there which contain at least one 7 or at least one 9…

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Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

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Q: Produce the least positive number that is divisible by​ 2, 3,​ 4, 5,​ 6, 7,​ 8, 9,​ 10, and 11.…

A: Given Produce the least positive number that is divisible 2, 3,​ 4, 5,​ 6, 7,​ 8, 9,​ 10, 11.…

Q: Calculate C(12,5) by hand. As C(n,r) = n!/(n-r)!r! identify n and r, and write out every number in…

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Q: the statement below true or false for n=1? ©George Mason University 2+ 4 + 6+ ... + 2n = n² + n + 1

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Q: For a natural number n, we define n! = 1·2·3· ... · (n − 1) · n. For example: 2! = 1 2,3! = 1 2 3 =…

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

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Q: determine the value of c(n,5) /p(n,5) where n can be any integer thats 5 or greater

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Q: 1. Show there exists an integer n such that n² + 3n/2 = 1. Is the integer unique? Explain your…

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Q: For an integer n > 1, how many decimal stings of length n begin or end with an even digit? 5(10)"-1…

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Q: For any integer n ≥ 1, C(6n, 3n) C(6n, 3n-1). O True O False

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Q: Shapley value of each senior = (31 + 3(1/2 + 1/2 + 1))/5040 = 37/840 Shapley value of each junior =…

A: Sure, I can explain where the numbers 31 and 30 come from in the calculation of the Shapley values

Q: The factorial of n is the product of all positive integers less than or equal to n. It is…

A: Note:Hey there! Thank you for the question! We have solved this problem using trial-and-error…

Q: Using Fermat’s little theorem to show that 42 | n7 −n, where n is a positive integer.

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Q: For every positive integer n     12+22+32+...+n2=n(n+1)(2n+1)/6 Let S(n)=12+22+32+...+n2 How do…

A: See the attachment

Q: How many triples of integers from 2 through 16 can be formed in which the integers are in increasing…

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Q: Derive an expression for the sum of the first n positive even integers.

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Q: Prove that a three-digit number is a multiple of 9 whenever the sum of the digits is a multiple of…

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Q: Jse a direct proof to show that if n² is an odd integer, then n* is also an odd integer.

A: Given, n² is an odd integer. So, n² = 2k+1, where k is an integer. Now, n⁴ = (n²)² = (2k+1)² = 4k² +…

Q: Prove that 8 | (54" +7) for every positive integer n.

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Q: 5) Let n be a positive integer such that 7 | (8-1). Prove that 7 | (8=*1 - 1). (Hint: Rewrite 8*1…

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Q: (1) Let S be the smallest set of positive integers such that (a) 2 is in S, (b) n is in S whenever…

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Q: integer of the form okts is also but not' conversely

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Q: Prove that 1 + 3 + 5 +···+ (2n − 1) = n2 for every positive integer n (a) by adding 1 + 3 + 5 +···+…

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Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

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Q: prove that the one's digit of any integer,n, is the remainder when one divides n by 10.

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Q: I also found that the expression was divisible by 13 at n=6 and n=10. Is there anyway you could…

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Q: If n is a positive integer, is there always a prime between n and 2n?

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Q: Use the approach in Gauss problem to find the sum of 8+17+26+35+...+890

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