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

Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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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