(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.

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A: Given that a0=6, an=3an-1     for n≥1.To show: an=6.3n   for all integers n≥0.

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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: os (sin e)de

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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: This problem is much harder than it looks. Use induction to show that n³ <3n Vn ≥ 1 Hint: you may…

A: For   So it is true.

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A: First step of Mathematical induction

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A: To Prove: 12+22+32+...+n2=nn+12n+16

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

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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: Guess a formula for 1+3+5+...+ (2n-1) for n ≥ 1 and prove your result by induction.

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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: 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: 5n + 3" > 22n+1 for all n E Z+

A: Here given as 5n+3n≥22n+1 for all n∈Z+ Put n=1 Then 5+3≥22+18≥8 It is true for n=1

Q: (4 – 2æ)".

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

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

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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

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A: Mathemetical Induction

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

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Q: Prove the following statement by induction. Theorem. Suppose n is a positive integer. Then n n…

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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: 8. Prove the statement below with either induction, strong induction or proof by smallest…

A: We can prove the statement  using the principle of mathematical induction.Base Case:Let n = 1.…

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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.