Problem 2: For positive integers n, r show that n+r-1) (^+ + + ¹) = (₁ + ²) + (" + ² + ¹) + " r r r-1 n +(+²)+(+¹)+() 1 n n+2 n+r- - (₁+¹)+(²+:-) +- + 2 n n+2) n + n n n + n n n

Q: Compute the following: (a) 4!, (b) (), (c) (₂), (d) (), (e) (7), (f) (2)-(2) where n is a…

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Q: 9(x + 6) - 41 = 75 Prove x - 62/9)

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Q: Prove that for all integers n.4n² +6n + 1 is not divisible by 4. Be sure to explicitly show how each…

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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: If C₁ * x² = c(1) ²¹ + a₂ (1) ²³ t² is the complementary solution of the nonhomogeneous system 3…

A: If the nonhomogeneous system is x'=3-22-2x+-2tt4-1 If the complimentary solution is…

Q: Please answer True or False for the following Statements... 1. For every odd integer n, the integer…

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Q: How can you prove or disprove: 3^n ≥ 2^n + 5n for all integers n ≥ 3?

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Q: Prove a^2 +4a+3 is odd if and only if a is even.

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Q: If In(a) = 2, In(b) = 3, and In(c) = 5, evaluate the following: %3D ) a-1 (a) In -12 b-3c-1 (b) In…

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Q: Prove the claim that there are no integers j and k such that j^2 + 3k^2 = 8

A: Since both j and k are being squared so if we prove the case for positive integers, that will prove…

Q: How many integer solutions to x1 + x2 + x3 + x4 x₁ ≥ 4, x2 ≥ 2, x3 ≥ 5 and x4 ≥ 3? > = 31 are there…

A: Find change all inequality to grater then equal to 0. Then use the theorem

Q: Prove that (a) For all integers m and n, n + m and n - m are either both odd or both even.

A: Since you have posted multiple questions, according to our guidelines we can solve only first…

Q: Prove the following using a direct proof.  If a,b, and c are integers such that b is a multiple of…

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Q: 10. If * -2 1 A -3 5 -1 3 Then cof (A) %3D -11 3 -5 -9 6. - 10 -3 1 11 -3 5 9 -6 10 3 -2 -1 None -11…

A: We have the matrix A=-210-3250-13 Now, if we have a matrix A=a11a12a13a21a22a23a31a32a33 then the…

Q: If X is the smallest, or first of three consecutive even integers express the sum of all three…

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Q: 1- x**! 1- x n+1 6 (x # 1) r=0

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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: Prove that m^3 + 2n^2 =36 has no solution in positive integers.

A: Prove that  there is no solution for positive integers.Let if for for

Q: Show that, for an integer n, if n^3 is odd, then n must be dd.

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Q: 38)-3x+17 or 6x + 32-21 A) [-4,-2) す B) (-4, ) C) (- ) D)(-2,)

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Q: 17. What Is the difference of 25 - -1 3x+25 (a+3)(x+5){z+2) » -3, x -5, e + -2 7z+25 O…

A: topic - subtraction of rational functions

Q: While proving (a*2(2a+1 is an integer for the case a = 6q +1 which of the following will be obtained…

A: We are given with a(a+1)(2a+1)6 We need to substitute a=6q+1

Q: n Prove that & (31²-2₁) = 2n³ tn ²-n 2=1 2 Goal: 2 (k+1) ³) + (1+₁)² = (K+1) 2

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Q: How many zeroes are at the end of 1000!/(500!500!)?

A: Calculate number of zeroes in 1000! and then subtract it from the number of zeroes in (500!500!).

Q: Prove that 3! 2(n+3) = n r=0 (-1) n "Cr r+3 Cr

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Q: prove the difference of the squares of two z integers is always divisible by 16 (i.e. z1^2 - z2^2 =…

A: We prove that (z_1)^2-(z_2)^2=16k for some integer k, then we say that divisible by 16.

Q: While proving n+2)2n+2) is an integer for the case n = 6q + 5 which of the following will be…

A: Given that  n(n+1)(2n+1)/6 Plug n=6q+5

Q: 10) (42x-33)+ (7x + 7) Please write in Q(x)+r(x)/d(x) Form

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Q: How do we know that -1 and -2 will be in the field?

A: Given F is field, So,  0, 1∈F ( since a field F must contain zero element and unity ) A field is…

Q: (4) For all natural numbers n, 4^n - 1 is a multiple of 3

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Q: 10. By Binomial Theorem (2a – b)* =16a* – 32a’b+24a*b? – 8ab³ +b* | a. TRUE. b. FALSE.

A: We know the binomial therom expansion

Q: Show that:

A: Given, nr+nr+1=n+1r+1

Q: (-5) + 4 – 1+ 4- (-21) * (-7)) ova

A: Simplification

Q: Prove by weak induction that (7^n) – 1 is a multiple of 6 fo

A: To prove that by weak induction 7n-1multiple of 6 for alln∈ℕ.

Q: If P(n) is the statement "n' + n is divisible by 3' Prove that P(3) is true but P(4) is not true.

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Q: Assume that n ∈ N. Prove that if 3|(2n + 1) then 9|(10n^ 2 + n − 2).

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Q: Show that n² #0 (4n²) by giving a simple argument

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Q: H) (3x² + 4x – x' – 2x' – 4) + (x+ 2) 0) (2x³ – 2x² + 4x² – 3) + (x + 1) P) (-x* + 2x – 2x - 3x + 1)…

A: we have to divide