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Let ao, a1, a2, ... be the sequence defined by the following recurrence relation: ao = 0 a, = aj-1 + i + 1 for i > 1 n(n+3) "rove that an = 2 for any nonnegative integer n.

Let ao, a1, a2, ... be the sequence defined by the following recurrence relation: ao = 0 a, = aj-1 + i + 1 for i > 1 n(n+3) "rove that an = 2 for any nonnegative integer n.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.4: Mathematical Induction
Problem 31E
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Part I: Induction Prove each of the following statements using induction, strong induction, or structural induction. For each statement, answer the following questions. а. Complete the basis step of the proof. b. What is the inductive hypothesis? C. What do you need to show in the inductive step of the proof? d. Complete the inductive step of the proof. 1. Prove that i · 2' = (n – 1) · 2"+1 + 2 for any positive integer n. 2. Prove that 11" – 7" is divisible by 4 for any positive integer n. 3. Prove that 3" > 2" + n² for any integer n > 2. Hint: Note that, for n 2 2, 2* > 1 and k² > k.
Transcribed Image Text:Part I: Induction Prove each of the following statements using induction, strong induction, or structural induction. For each statement, answer the following questions. а. Complete the basis step of the proof. b. What is the inductive hypothesis? C. What do you need to show in the inductive step of the proof? d. Complete the inductive step of the proof. 1. Prove that i · 2' = (n – 1) · 2"+1 + 2 for any positive integer n. 2. Prove that 11" – 7" is divisible by 4 for any positive integer n. 3. Prove that 3" > 2" + n² for any integer n > 2. Hint: Note that, for n 2 2, 2* > 1 and k² > k.
4. Let ao, a1, az, .. be the sequence defined by the following recurrence relation: ao = 0 a, = aq-1 + i +1 for i >1 Prove that an n(n+3) for any nonnegative integer n. 2 5. Let bo, b1, b2, .. be the sequence defined by the following recurrence relation: bo = 51 b1 = 348 bị = 5· bi-1 - 6 · b¡-2 + 20 · 7' for i > 2 Prove that b, = 2" + 3" + 7"+2 for any nonnegative integer n. 6. Let c,, C2, C3, . be the sequence defined by the following recurrence relation: C1 = C2 = C3 = 1 Ci = Ci-1+ Cj-2 + Ci-3 for i > 4 Prove that c, < 2" for any positive integer n.
Transcribed Image Text:4. Let ao, a1, az, .. be the sequence defined by the following recurrence relation: ao = 0 a, = aq-1 + i +1 for i >1 Prove that an n(n+3) for any nonnegative integer n. 2 5. Let bo, b1, b2, .. be the sequence defined by the following recurrence relation: bo = 51 b1 = 348 bị = 5· bi-1 - 6 · b¡-2 + 20 · 7' for i > 2 Prove that b, = 2" + 3" + 7"+2 for any nonnegative integer n. 6. Let c,, C2, C3, . be the sequence defined by the following recurrence relation: C1 = C2 = C3 = 1 Ci = Ci-1+ Cj-2 + Ci-3 for i > 4 Prove that c, < 2" for any positive integer n.
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