a1 = 1, an+1 = 3 - An induction method to prove monotor

4. Let P(n) be the statement that a postage of ʼn cents can be formed using just 4-cent stamps and 7-cent stamps. The parts Page 363 of this exercise outline a strong induction proof that P(n) is true for all integers n ≥ 18. a) Show that the statements P(18), P(19), P(20), and P(21) are true, completing the basis step of a proof by strong induction that P(n) is true for all integers n ≥ 18. b) What is the inductive hypothesis of a proof by strong induction that P(n) is true for all integers n ≥ 18? c) What do you need to prove in the inductive step of a proof that P(n) is true for all integers n ≥ 18? d) Complete the inductive step for k ≥ 21. e) Explain why these steps show that P(n) is true for all integers n ≥ 18.

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Let S be the subset of the set of ordered pairs of integers defined recursively by Basis step: (0,0) = S F Recursive step: If (a, b) = S, then (a, b + 1) = S, (a + 1, b + 1) = S, and (a + 2, b + 1) = S. List the elements of S produced by the first four applications of the recursive definition. Enter your answers in the form (a₁, b₁), (a2, b2),..., (an, bn), in order of increasing a, without any spaces. The first application of the recursive step adds (Click to select) ✓to S. The second application of the recursive step adds (Click to select) The third application of the recursive step adds (Click to select) The fourth application of the recursive step adds (Click to select) to S. ✓to S. ✓to S.

Use strong induction to prove that √2 is irrational

Exercise 1. (a) Determine the smallest value of a for which all natural numbers n ≥ a, n! > 3n File Preview (b) Prove using mathematical induction that for natural numbers n ≥ a, n! > 3n.