Theorem: If n is an odd integer, then ⌊n/2⌋ = (n-1)/2.

Q: 20. (a) Show that the sum of two odd prime numbers is never prime. (b) Is (a) true if the word odd…

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Q: n(n+1)(2n+1) 6 In the proof that the sum of the first n squares is we showed P(0) and then P(k) ⇒…

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Q: 2(5) – 3(1 – 6) 1. Is even integer? 5 2. List all even integers between -6 and 6. 3. Is 124/6…

A: Four questions are given. The solution to the four questions are given below.

Q: (n-1)! Use Euler's theorem to prove that if n is an odd integer, then n divides 2 -1.

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Q: (a) Please fill out the rest of the table. n(n+1)² n² (n + 1)² -n² -3 -2 TOI231 0 4 5 61490 4 1 0 1…

A: (a) The complete table will be as follows: n (n+1)² n² (n+1)² - n² -3 4 9 -5 -2 1 4 -3 -1 0…

Q: (1+ 2x + 4x + 8x° + · · · )(3x + 3x* + 3x° + · · · )(x° + 2x° + 3x° + •..)

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Q: (g) 111... 111 (126 ones) (2) Prove that the numbers given in each part cannot be prime…

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Q: Let r be a real number and let k be an integer. We then define the binomial coefficient () by r(r-1)…

A: Given, r is a real number and k is an integer  The Binomial Coefficient rk by rk= r (r - 1) --- (r -…

Q: (a) Prove that for every integer n ≥ 2, 1 n+2 1 + n+1 +...+ 2n

A: Since you have posted a multiple question according to guildlines I will solve first (a)question for…

Q: Зх —1 A B (b) (i) Express (1– x)(2 - 3x) in the form - x 2-3x , where A and B are integers. Зх— 1…

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Q: ) 6. Express the sum 9 + 28 + 65 + ... + (n'+1) using E-notation.

A: We have to find sum of series:

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: Prove that n²– 3n + 1 is odd for every integer n.

A: To prove that n^2-3n+1 is odd for every integer n.

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

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Q: Find the largest integer k for which fl(19+2-k) > fl(19) in double precision floating point…

A: Advance maths

Q: 20. (a) Fill in the blank: 1+3+5+..+ (2n – 1) = k=1 (b) Use part (a) to prove that the sum of the…

A: This is an arithmetic series with common difference (3-1)=2 And this has n terms in it So we use the…

Q: 3. Prove each of these divisibility statements by mathematical induction. (b) (a) 8|52n +7 for all n…

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Q: Question 12 Which of the following is not correct? B n²(n+1)(n+2) 3 is 0 (n) for every integer n ≥ 3…

A: (A) n2n+1n+23 is θn4 for every integer n≥3     ∴n2n+1n+23=n2n1+1nn1+2n3=n431+1n1+2n=θn4 Hence option…

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

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Q: (a) There exists two odd integers whose average is odd. 1 1 (b) There exists distinct positive…

A: (a) ans:- 1,5 (b) ans:- 2,3 &6 For detail...go through the solution

Q: Prove that 1¹x 2² x 3³ x...xn" < n(n+1) 2 (2n+1) 3

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Q: (b) Show that 1 11 2 k(k + 2)(k + 4) 96 k=1

A: Given series is ∑k=1∞1kk+2k+4

Q: Verify that for all n ≥ 1, the sum of the squares of the first 2n positive integers equals…

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Q: (a) Prove that for all integer n 2 1, 3|n(n² + 11).

A: a.          In the given problem we have to prove that for all integer n≥1, 3|n(n2+11). To prove…

Q: A positive integer n is called superperfect if ơ(ơ(n)) = 2n Show that if n = 2q, where 2q+1 - 1 is…

A: Given: A positive integer n is called superperfect if σσn=2n and n=2q, where 2q+1-1 is prime. To…

Q: For any positive integer n, prove that 6|n(7n² + 5).

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Q: umbers and q (> 0) the first of n harmonic means between the me numbers. Then prove that

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Q: (3n)! is an integer for all n > 0. (3!)n Prove that

A: We will prove it by Mathematical induction. First We show it is true for initial condition n=0…

Q: n5 7n Prove that 3 is an integer for n > 1. 15

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Q: n²+3 4 Prove that if n is an odd integer, then √2-2

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Q: n(n?+5) is divisible by 6, for each each integer n20 Proof (by induction) P (0) = " o (0²+5) is…

A: In this question, concept of mathematical induction is used. Mathematical Induction Mathematical…

Q: -1/2 p. Find the coefficient of x" in the expansion of (1– 4x)2.

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