[C] Prove: For all positive integers k, if the sequence sk : (k− 1)(k), k(k), (k+1)(k) is graphical, then k is even.

Q: Prove that (2k+1)! > (k+1)!(k+1)k

A: We shall prove this by induction on k. For initial step, put k=1. LHS=2×1+1!=3!=6 RHS=1+1!1+11=4 LHS…

Q: The Lucas numbers L(n) have almost the same definition as the Fibonacci numbers: if n = 1 if n = 2…

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Q: To show that the general factorial function: (a)= mn (-). (+¹).(+-¹). here m is a positive integer…

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Q: (a) For every integer n, the number n² — n must be even. (b) For all real numbers a, a² < a¹. (c)…

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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: Given that f(n) is a function for all non-negative integers n, find f(2), f(3), and f(4) for each of…

A: topic - functions

Q: Show that 1+2+3+… +n = (";").

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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: Use a combinatorial argument to prove the following equality: (C) · k = =n(") k-

A: A combinatorial argument or proof is an argument that involves counting. Here is the proof.

Q: (2) Translate the following English sentences into symbols. (a) Every real number has a cube root.…

A: Please go through the solution posted below.And If you have any doubts then please ask me in the…

Q: (c) Prove that for n ≥ 1, n [27/7/2≤2- j² 1 n

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Q: Find all integers n such that o(n) = 40.

A: Use the multiplicative property of ϕ:  if n and m are relatively prime then  ϕ(nm) = ϕ(n)ϕ(m). 40  =…

Q: For positive integers n and k (k sn+ 1), the kth term of (a + b)n is given by

A: The binomial expansion of (a+b)^n is

Q: K (3x + 16)° (11x- 18) M

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Q: that for n > k we Sn+1 ||

A: The expression given is ,  n+1k+1=∑j=knnn-jjk The number of ways to partition n+1 objects into k+1…

Q: For all integers n>0, prove: 2n < (n+2)!

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Q: Use the Geometric Sum identity to prove that, if r 70, then (1 – x)3 1 =1+ (1 - x) + (1 – x)² + |

A: The objectives to use geometry sum identity to prove the following expression.

Q: Prove that k (x)= - = n (n − 1) for integers n and k with 1 ≤ k ≤ n, using a: a) algebraic proof.

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Q: Prove that (n+1)C n is equal to n+1

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Q: Prove:  C(n,k) = C(n-1,k-1)+C(n-1,k) for natural numbers k with 0<k<n.

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Q: (a) Find the product a b, given that a = b = √9-√√9-√9-√7. b) Find all sequences a1, a2, a3,....of…

A: as per company guidelines we solve only first question when multiple question being posted

Q: Evaluate |P(A)| if A={n: n is a prime number from 1 to 100}

A: To evaluate : PA if A=n : n is a prime number from 1 to 100

Q: Define P(n) to be the assertion that: n(n + 1)(2n + 1) 6 j= (a) Verify that P(3) is true. (b)…

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Q: In the inductive step, we assume that for any k > 10 , P(j) is true for any 8 <js (k-2) and we will…

A: The proof by mathematical induction consists of two steps: The proof consists of two steps: The…

Q: Show that for any integer n ≥ 2, C(2n,n) > C(2n,n-1)

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Q: (c) 17. The smallest positive integer n for which (E) = 1 is... (d) (a) 8 (b) 16 (c) 12 (d) 4. (e)…

A: complex numbers

Q: For each integer n>1, let A(n) denote the sum of the integers from 1 to n. For example,…

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Q: (2) (a) Prove by contrapositive : Va, b e R* : a <b → a < a +b < b. (b) Prove by contradiction : For…

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Q: (a) Prove that x, y = -1, y = 1 1 (b) Prove tha and n: m and

A: This is a problem of Number Theory.

Q: Let f(n) = 5" – 4. (a) Evaluate f when n is 1, 2, 3, 4, and 5. (b) For what positive integers n is…

A: The function is  :

Q: . Let f(n) be defined recursively by f(0) = 3, and f(n + 1) = 3f(n)/3, for n = 0, 1, 2, . . . Then,…

A: The recursive definition f(n+1)=3f(n)/3 introduces a function that relies on its own values to…

Q: (c) For every positive integer n, prove that the number of positive integers c with n<√e+ √e< n + 1…

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