Prove that: (1+ x)*** (n+ 1)x C. + + 2 3 n+1

Q: _If A = Verify that [A-!]" = [A"]¬1 ? l1 2.

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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: Prove n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

A: We have to proff the question

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: Find Im(z) if z(8 + i) = 7+ (384 + 448i)

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Q: In(x²) = 2 ln(x) for all x True O False

A: ln(x^2) =2ln(x) for all x

Q: In (5 – x) – In (5 + x) = -In 9

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Q: The following is a valid identity, true or false? P(k): 1+2+22 +23 + .. +2k = (2(k+1)/2) +…

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Q: Prove 3 | 17k+1-2*5k+3+11k+3 for k>=0 (nonnegative integer)

A: Here in this problem we have to prove that for k≥0 i.e k is a non negative integer…

Q: greatest common divisor(g.c.d)

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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: (c) Prove that r| +11 36 T-1 2 +36 for all real I.

A: We have to prove that the following inequality holds: x-11x2+36≤x+1136 for all real x. We will use…

Q: If P (n) is the statement "n + n" is even", and if P(r) is true, then prove P (r + 1) is true.

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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: Find the smallest positive integer in the given set. (a) {6u + 15v : u, v ∈Z}    (b) {12r + 17s : r,…

A: Given: (a) { 6u+15v:u,v∈Z }(b) { 12r+17s:r,s∈Z }

Q: Maximise Z = 5x1 + 6x2 Subject to X1 + x2 0 and integer

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Q: (1) As usual, let (a, b) stand for the Greatest Common Divisor of a and b. Is it true that k(a, b) =…

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Q: 11. Prove that every positive integer n can be expressed in the form n=4k or n=4k+1 or n=4k+2 or…

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Q: If the sum of the coefficients of all even powers of x in the product (1 + x + x² + ... + x2n) (1 –…

A: Introduction :

Q: Show that if s(3) 4(q- _9−2+9(1+ 93 2(g-1) increasing or decreasing. For 3 > 0, q> 3

A: The given function is                               sβ=q-2+q1+4q-1qβ122q-1 where β>0 and q>3.…

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

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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: 1.5 Inx - In(x2 - 1) + 2ln(x + 3)

A: Logarithm simplification

Q: Which of the following statement is always false? (a) For all integer x, [x] = [-x] (b) For all odd…

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Q: Show that if n2 -3 is even, then n is odd. Prove by contrapositive For the toolbar, press ALT+F10…

A: We need to prove by contrapositive.  The contrapositive statement of "If p then q" is "If not q then…

Q: Prove the property of the Fibonacci numbers, F(n) = 5(n 4) + 3F(n-5), for n ≥6

A: We have to prove the property of the given Fibonacci numbers F(n) = 5F(n-4) + 3F(n-5) ,  for n≥6

Q: Use the method of direct proof to prove that if x is an even integer,  then x² is even

A: Assumption: Let x be an even integer.By the definition of an even integer, we can express x as 2k,…

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

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Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: Compute: 3] 2 5-3

A: 1.a1a2a3a4+b1b2b3b4=a1+b1a2+b2a3+b3a4+b42. Two matrices can be added iff the order of the marices…

Q: Find the general solution of the recursive relation an = 2 an-1 - 6 an-2 for every integer n > 1. a,…

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