a) Show by example that (3) in general is not equal to . And Х b) it is possible for x to occur in () instead of where it does not occur in 4, such as x x occurs in instead of where it does not occur in (3). c) Show that if y does not occur in §, then x is substitutable for y in på and (x) = p. Hint: Use induction on . x

Q: Show that for all n> 14, n cents can be represented by some combination of 3-cent coins and 8-cent…

A:

Q: 3. Show by mathematical induction that, for all n2 0, the equality +++.+ = 2 – +2 holds. ... 2Th

A:

Q: Let n≥1, x be a real number, and x ≥ −1. Prove the following statement using mathematical induction.…

A:

Q: Let x, y and n denote integers. Use induction to prove that Vn 2 5: 3x,y 20 such that n = 2x + 3y

A: Strong induction: Let p(n) be a statement . Then (i) let p(n) is true for a base case p=m. (ii) If…

Q: Use mathematical induction to show that E"-, ar ark-1 a(1-r") for all integers n 2 1-r 1. (Assume r…

A:

Q: 2. Let Q(n) be the statement Suppose “P(1), P(2),..., P(n) are all true. 22 • P(1) is true; ● for…

A:

Q: 3. Let a, be an odd integer for cach integer n > 1. Prove that if n is odd, then the sum Ea, is odd…

A:

Q: prove for all ne J, 1+ 3 + 5 +..+(2n-1) = n² by induction

A: The statement is for all To prove :The above statement is true using mathematical induction.

Q: Let fn be the n-th Fibonacci number of Example 3c in 8.1, k=n k=n = An Σ=1+2+. .+ n₂ =k²= 1² +2²+.…

A:

Q: Prove that in a Boolean Algebra, x.ỹ = 0 if and only if x. y = x .

A:

Q: Let A, B₁, B, be sets. Using induction on 2, prove the following statements: (a) AU (B₁n B₂n CHICA…

A:

Q: Prove using induction that every positive even integer can be written in the form 2Pm, where p E N…

A:

Q: (b) Prove that for any positive integer n, Ej. 2 = (n – 1)2"+1 + 2 (c) n(n² – 1) Prove that for any…

A: b) Let the statement be P(n): 1·21+2·22+3·23+⋯ +n·2n(, or,∑j=1nj·2j )=(n-1)2n+1+2. For n=1, P(1) :…

Q: By Principle of Mathematical Induction prove 6n 6

A:

Q: (b) Prove that for any positive integer n, j. 2³ = (n – 1)2"+1 +2 j=1 (c) n(n? – 1) - Prove that for…

A:

Q: Use Mathematical Induction to prove

A: Initial case:

Q: Prove the following statements by induction. Note that n is a positive integer. (rn+1_1) 1.1. E-o rk…

A: As per our company guidelines we are supposed to answer only the first question. Kindly re post the…

Q: 7 Use mathematical induction fo derive the following formeld for gll nzi!

A:

Q: Let n be an element in Z. Prove that 5|n^2 if and only if 5|n.

A: If n be an element in Z. We have to prove that 5|n^2 if and only if 5|n.

Q: (e) Prove that for any non-negative integer n, i 3 n - 3"+1 – (n+ 1)3" + 1] 4 j=0 (f) Prove that for…

A: (e)…

Q: prove f(x)=2x (mod 7) is injective

A: To prove that the function f(x) = 2x (mod 7) is injective, we need to show that for any two distinct…

Q: 1 Let x € (0, 1). Use induction to prove that xn for all natural numbers n ≥ 2. X

A:

Q: Prove that 36n is not O(35n)

A: Let, 36n∈O35n

Q: a(1–r") for every 1-r Let r + 1 be a real number. Use induction to prove that a + ar + ar² + . ..+…

A:

Q: Q1. Let x and y be two integers. Prove that for all integers m 2 1, x™ – y™ is a multiple of x-y…

A:

Q: EXERCISE 4.11.2 ONLY DO (D-F)****

A: For part d: According to the given information it is required to prove that for any positive integer…

Q: 5.2.8 More generally, use induction to prove that (a - b) | (a" – b") for any positive integers…

A: Given: (a-b) | (an-bn) ----(1) Substitute n=1 in equation (1): (a-b) | (a1-b1)=1 The assertion is…

Q: Prove using mathematical induction that for all n > 1, n(3n – 1) 1+4+7+·..+ (3n – 2) 2

A:

Q: Use mathmatical induction to prove that for all intergers n is greater than or equal to 1.…

A: For n = 1,       LHS = 4 [only 1 term]      RHS = 2.12 + 2.1 = 2.1 + 2.1 = 4 Hence, LHS = RHS.  So,…

Q: Let a₁ = a₁ = a₂ = 2 and an = an-1 + an-2 + an-3 for n > 3. Prove, using strong ao mathematical…

A: