(d) Define the absolute value of a E R. Hence or otherwise find the value of |(-1)"+1 – (-1)"+2| (e) Let a,, n 2 1 be a sequence of natural numbers such that a,41 > a, nEN. Use induction to show that a,n 2 n, V n E N.

Q: Use induction to show that Vn EN, 6| 7n – 1.

A: For the solution follow the next step.

Q: 3. Use the Principle of Mathematical Induction to prove that for a natural number n, n(n + 1)(2n +…

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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: (d) Define the absolute value of a E R. Hence or otherwise find the value of |(-1)"+1 – (-1)"+2| (e)…

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Q: The sum + is between 8 (A) and 3 (B) and 1 (C) 1 and 1 14/ D) 1 and 1/2 E) 1 and 2

A: The given is  Sum of this can be calculated  as           =                         =…

Q: 2. The Fibonacci numbers F(0), F(1), F(2),… are defined as follows: F(0) := 0 F(1) := 1 F(n) := F(n…

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Q: Use Mathematical Induction or Strong Mathematical Induction to prove the given statements (a). V n €…

A: To prove a statement by Mathematical induction method, we have show that given statement is true for…

Q: Assume that you are asked to use mathematical induction to prove a propositional function P(n) is…

A: First step of Mathematical induction

Q: Prove by induction. 3F, = Fn+2+ Fn-2 for n > 2

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Q: product 16: n! = 1.2.3.... · (n − 1) · n. = 120. example, 5! = 1.2.3.4.5 e induction to show that…

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Q: Use induction to show that Vn > 2, 6| 7" – 1.

A:

Q: 1. Use mathematical induction or strong mathematical induction to prove the given statement. 1 (a).…

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Q: 11. a) Find a formula for 1 +... + 2" by examining the values of this expression for small values of…

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Q: 9) Let a, = a-1 +12a__2,a, = 2, a, = 1.Use the strong induction to show that for all n20, a, = (-3)"…

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Q: n Prove by induction that Σ (− 1) ¹;² = (-1)”n(n+1) 2 i=1 n Prove that Σ i=1 1 i(i+1) = n n+1

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Q: Prove by induction the identity (n + 1) · (n + 2)... (2n − 1) · 2n 1.3.5.... (2n − 1) for any…

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Q: (b) Prove by induction for all n >1 that an – bn = 2bn-1· %3D

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Q: et r +1 be a real number. Use induction to show that for any positive integer n, a(rn+1 – 1) a + ar…

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Q: (3c) Let n E Z. Prove that if n is odd, then 4|(n² – 1)

A: here is the proof

Q: Finish this proof by mathematical induction by completing the missing steps 3 and 4 Prove…

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Q: Use Mathematical Induction to show that, for all n € Z†, 6 | (10″ + 2).

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Q: Let m, n be odd integers. Prove that m +n is even. Remember: An integer b is odd iff 3k. (kE ZA 2k +…

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Q: Find the dual of the Boolean expression I +1.0+ Fyz O z. (z+1). (7+ y+z) O T. (7 +0). (7 + y+z) O z.…

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Q: Q2. Use induction to prove that for any natural number n , 2-1 = 2 – 2-".

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Q: Ź K=n*(n+1) 4 出2 Use induction to Show

A: To use mathematical induction to prove the given statement.

Q: r² – rn+1 5. Let r +1 be a real number. Use induction to show that r* for all 1-r k-2 ne Z+ with n >…

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Q: (e) Show that every n ≥ 8 can be written as 3a+5b for some a, b € N.

A: Part e asked and answered.

Q: (a). For all integers n ≥ 8, there exist integers l and m such that n = 3l + 5m.

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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: (1) Prove by induction: Σ²₁ (i² − i + 1) = n³+2n

A: (1)  Let the given statement is  P n :  ∑i=1n i2 - i + 1 = n3 + 2n3             For  n = 1  we get…

Q: Consider the Fibonacci numbers {F, = 1, F2 = 1, F3 = 2, F4 = 3, Fs = 5, F6 = 8,· ·}. Prove by…

A: I have given the proof in second step

Q: Use proof by induction to prove that: =-n°(n+ 1)? %3D r=1

A: Mathemetical Induction

Q: Use mathematical induction to show that, Vn > 3, 2n² + 1 > 5n.

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Q: 4. Use Mathematical Induction to prove: (a) Vn € N, 2+5+8++ (3n − 1) = n(3n+1) 2

A: Querstion No. 4(a): By using mathematical induction to propve the following statement. Note: Since…

Q: (c) Show that for 2" > 4n for n ≥ 5.

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Q: Use the mathematical induction

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Q: 1. Use Mathematical Induction or Strong Mathematical Induction to prove the given statements. (a).…

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.