Prove the following statement using mathematical induction. Do not derive it from Theorem 5.2.1 or Theorem 5.2.2.For every integer n2 1, 1 + 6 + 11 + 16 + ... + (5n - 4) = n(5n - 3)Proof (by mathematical induction): Let P(n) be the equation1+6+ 11 + 16 + ... + (5n - 4) = n(5n - 3)2We will show that P(n) is true for every integer n2 1.Show that P(1) is true: Select P(1) from the choices below.O P(1) = 5. 1 - 4O 1 =1:(5 1- 3)2O 1+ (5.1- 4) = 1. (5. 1- 3)O P(1) = 1: (5.1- 3)2The selected statement is true because both sides of the equation equalShow that for each integer k2 1, if P(k) is true, then P(k + 1) is true:Let k be any integer with k 2 1, and suppose that P(k) is true. The left-hand side of P(k) is ---Select---v, and the right-hand side of P(k) is[The inductive hypothesis states that the two sides of P(k) are equal.]We must show that P(k + 1) is true. P(k + 1) is the equation 1+ 6 + 11 + 16 + --- + (5(k + 1) - 4) =. After substitution from the inductive hypothesis, the left-hand side of P(k + 1) becomesSelect--v + (5(k + 1) - 4). When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equalHence P(k + 1) is true, which completes the inductive step.[Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]

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Description: Prove the following statement using mathematical induction. Do not derive it from Theorem 5.2.1 or Theorem 5.2.2.For every integer n2 1, 1 + 6 + 11 + 16 + ... + (5n - 4) = n(5n - 3)Proof (by mathematical induction): Let P(n) be the equation1+6+ 11 +

For, n=1, 1=1(5×1-3)2=1 This statement is true because both sides of the equation are equal.

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rove the following statement using mathematical induction. Do not derive it from Theorem 5.2.1 or Theorem 5.2.2. n(5n - 3) For every integer n2 1, 1 + 6 + 11 + 16 +... + (5n - 4) = 2

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.3: Divisibility

Problem 49E: 49. a. The binomial coefficients are defined in Exercise of Section. Use induction on to prove...

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