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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 4,5,6 only please
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.
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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