1. Induction and prime factorisation of natural numbers: Let n EN be a natural number. A prime factorisation of n is an equation n = p'p...p , where for i = 1,... , k, pi is a prime number and d; E {0}UN. Using Mathematical Induction, prove that every natural number has a prime factorisation. di, d2 de The following statements inay be useful in your proof: (Q1) Vm, n €N (m < n) = < : (Q2) Vm, n E N (m and n have prime factorisations) (mn has a prinie factorisation); (Q3) Vn E N (n > 2) =

Q: 7n is a 15 n° Use mathematical induction to show that 5 3 natural number for all n E N. + +

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Q: H2) Prove, using mathematical induction, that 32N-1 + 1 is divisible by 4 for any NE N. (Hint: A…

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Q: Use induction to show that the following identity holds for all natural numbers: (n+1)(5n+6) Eo (3+…

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Q: 1. Use the principle of mathematical induction to prove that 1 +r+r² + ³ + ... + pn = 11 pn+1 - r-1

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Q: * 63. Let a¡, az» ·… , A, be positive real numbers. The arith- metic mean of these numbers is…

A: The arithmetic mean AM=a1+a2+...+annThe geometric mean GM=(a1,a2...,am)1n We need to show A≥G.

Q: The first few powers of 9 are 90 = 1, 91 = 9, 92 = 81, 93 = 729. Prove by strong induction that 9n…

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

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

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Q: For ne N, 2n -8 < n²-8n + 17.

A: 3.7 We have to prove that 2n−8&lt;n2−8n+17 for all n∈N by principle of induction method. Let n=1…

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: n, that Fo+F1+F2+· ··+Fn = Fn+2-1, number (Fo = 0, F1 1 and Fn

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Q: Prove the following using Mathematical Induction, for n>1 7" 1 is divisible by 6

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Q: Prove the following using Mathematical Induction, for n>1: 8 - 3" is divisible by 5

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Q: O A. No, since we have only shown that Sk+ 1 satisfies the statement. OB. No, since we must show…

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Q: Let S be a subset of integers defined recursively as follows: Basis step: 2 € S. Recursive step: if…

A: Given- Let S be a subset of integers defined recursively as follows: Basic Step: 2 ∈ S Recursive…

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: Let aj = 1 and 1 -ап, п >1. n+1 аn+1 Show by mathematical induction that for all positive integers…

A: we have to solve by using mathematical induction

Q: Use mathmatical induction to prove that the statement is true for every positive integer n.

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.