2 4) The graphs of y=x and y=x make it obvious that x >x for all real number x > 1. Use induction (not graphs) to prove that 3 3 2 n >n for all integers n > 1.

Q: 2 4) Prove that n + 5 > 4 n for all integers, n >0.

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Q: show that for all integers x and y (x-y)3=x3-y3(mod3)

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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: Suppose that n pigeons are placed into m holes with m ≥ k(n+ 1) for some positive integer k. Suppose…

A: See the attachment

Q: e3x e(n+1)x = 1+ e" + 2 enx Sx 6. n! (n + 1)!

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Q: Use the principle of mathematical induction to prove | (In x)" da = (-1)" - n!

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Q: -7 3. (15 points) Use mathematical induction to prove that II (211 II (21+1 2+3)-(2 + 2) for all…

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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: Using the generating function to determine the number of ways  18 identical balloons can be given…

A: Given that 18 identical balloons can be given to 3 children such that each child receives at least 5…

Q: Show that 1+2+3+… +n = (";").

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Q: Prove that for all positive integers, 1 1 + 3.4 1 ... 1.2 2-3 n(n + 1) n+1'

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Q: 10. Use induction to show that 1x2 + 2x3 + + n(n+1) = " for all n € 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: 5. (20) Prove by induction that for positive integers n, 4|(5"+1 + 32n-1).

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Q: Prove by induction that for positive integers n: E (312 + 2i + 5) = }(2n³ + 5n² + 13n)

A: To prove the given statement by mathematical induction.

Q: 13. Use mathematical induction to show that 1+4 4-1/344-1).

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Q: 7. Ug = kuz-1¬- Ug–2, for every integer k > 3 uj = 1, u2 = 1

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Q: Prove that for all natural numbers k, a (5*2^(2k+1) - 1)/3 + a*4^k is of the form 5 + 6a.

A: The question looks incorrect and cannot be proved. This is because of the following.…

Q: 12 Let u₁ = 1, ₂ = 1 and un + 2 = Un + 1 + un n ≥ 1. Then use mathematical induction to show that 1…

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Q: Consider certain real numbers ui and vi, i = 1,..., n, n <∞0. Show that (u₁v₁ + U₂v₂)² < (u² + u²)…

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Q: The multiplication rule says that if an operation can be performed in k steps and, for each i with 1…

A: Answer is mentioned below

Q: 5) Let a, = 2a +5,a, =1, use induction to show that a = 6(2")-5 for all n20.

A: Given that an=2an-1+5 and a0=1. We have to show that an=62n-5 for all n≥0. We have to use the…

Q: Let P(n) be the statement that 1^2+2^3+· · ·+n(n + 1) = n(n + 1)(n + 2)/3 for the positive integer n…

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Q: 1) Let x be a positive real number. Show by induction on n that (1+x)" > 1+ nx for all integers n >…

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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: 4) The graphs of y=x and y=x make it obvious that x >x for all real number x > 1. Use induction (not…

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Q: Find the smallest integer i for which the statement (1 + i2 )<2' is true. Then, prove the statement…

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Q: Show by induction that for each positive integer k, 1?+ 2² + 3² + . . · +k² = }k(k+ 1)(2k + 1).

A: NOTE: Refresh your page if you can't see any equations. . check for k=1 the statement is true…

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: 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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Q: Give a combinatorial proof for the identity 1+2+3+..+n= (",").

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Q: Define 0! = 1. For any positive integer n, define n! = n- (n- 1)!. For example, 6! = 6x5x4x3x2×1=…

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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: 3) Let the order on the X=(-∞0,5√2) NQ be the known order. find the supremum Supremum and infimum of…

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