Question 4,5,6 only please Part I: InductionProve 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 thati · 2' = (n – 1) · 2"+1 + 2for 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 = 0a, = aq-1 + i +1 for i >1Prove that ann(n+3)for any nonnegative integer n.25. Let bo, b1, b2, .. be the sequence defined by the following recurrence relation:bo = 51b1 = 348bị = 5· bi-1 - 6 · b¡-2 + 20 · 7' for i > 2Prove 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 = 1Ci = Ci-1+ Cj-2 + Ci-3 for i > 4Prove that c, < 2" for any positive integer n.

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Answered: Let ao, a1, a2, ... be the sequence…

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

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.4: Mathematical Induction

Problem 42E

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Question

Question 4,5,6 only please

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.

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