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Need to label the inductive hypothesis. what would be the inductive hypothesis

Need to label the inductive hypothesis. what would be the inductive hypothesis

Advanced Engineering Mathematics
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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Need to label the inductive hypothesis.

what would be the inductive hypothesis?

Basis step: By the definition of the sequence bị = 4 and b2 12 implies that 4|4 and 4|12. So, P (1) and P (2) are true for n = 1,2. Inductive step: Show that for all integers k > 1, if the property P (k) is true then P (k + 1) is true. For that, Assume that the result is true for n = k and k > 3. Also br = bk-2 + bk-1 That is 4|bx = 4| (bx-2 + b-1) = 4|bx-2 + 4|bx-1 Therefore, form the assumption 4|bx, 4|bk-2, and 4|bg-1 for all k 2 3.. (1). Need to show that P (k + 1) is true. For that it is enough to show that 4|b+1. k+1•
Transcribed Image Text:Basis step: By the definition of the sequence bị = 4 and b2 12 implies that 4|4 and 4|12. So, P (1) and P (2) are true for n = 1,2. Inductive step: Show that for all integers k > 1, if the property P (k) is true then P (k + 1) is true. For that, Assume that the result is true for n = k and k > 3. Also br = bk-2 + bk-1 That is 4|bx = 4| (bx-2 + b-1) = 4|bx-2 + 4|bx-1 Therefore, form the assumption 4|bx, 4|bk-2, and 4|bg-1 for all k 2 3.. (1). Need to show that P (k + 1) is true. For that it is enough to show that 4|b+1. k+1•
Given information: Suppose b1, b2, b3, ... is a sequence defined as follows. bị = 4, b2 = 12, bk = bk-2 + bk-1, for each integer k > 3. Concept used: Let P (n) be the sentence 4|bn . By the definition of the sequence bị 4 and b2 12 implies that 4|4 and 4|12. So, P (1) and P (2) are true for n = 1,2. Proof: Suppose b1, b2, b3, . is a sequence defined as follows; ... ..... bị = 4, b2 = 12, bk = bk-2 + bk-1 for each integer k > 3. The objective is to prove that b, is divisible by 4 for all integers n > 1.
Transcribed Image Text:Given information: Suppose b1, b2, b3, ... is a sequence defined as follows. bị = 4, b2 = 12, bk = bk-2 + bk-1, for each integer k > 3. Concept used: Let P (n) be the sentence 4|bn . By the definition of the sequence bị 4 and b2 12 implies that 4|4 and 4|12. So, P (1) and P (2) are true for n = 1,2. Proof: Suppose b1, b2, b3, . is a sequence defined as follows; ... ..... bị = 4, b2 = 12, bk = bk-2 + bk-1 for each integer k > 3. The objective is to prove that b, is divisible by 4 for all integers n > 1.
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