By dragging statements from the left column to the right column below, give a proof by induction of the following statement: an = 7 - 1 is a solution to the recurrence relation an = 7an-1 +6 with a = 0. The correct proof will use 8 of the statements below. Statements to choose from: Your Proof: Put chosen statements in order in this
By dragging statements from the left column to the right column below, give a proof by induction of the following statement: an = 7 - 1 is a solution to the recurrence relation an = 7an-1 +6 with a = 0. The correct proof will use 8 of the statements below. Statements to choose from: Your Proof: Put chosen statements in order in this
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
Transcribed Image Text:Then 7k+1 - 1 = 7(7k . − 1) + 6, which is true.
Now assume that P(k + 1) is true.
This simplifies to
=
: 7k+¹ − 7 + 6 = 7k+1
-
Thus P(k+ 1) is true.
Therefore, by the Principle of Mathematical
Induction, P(n) is true for all n ≥ 1.
Now assume that P(n) is true for all n ≥ 0.
Let P(n) be the statement, "an = 7" - 1 is a
solution to the recurrence relation
an = 7an-1 +6 with a = 0."
Thus P(k) is true for all k.
Then = 7¹ - 1.
ak+1
1
By the recurrence relation, we have
ak+1 = 7ak + 6 = 7(7⁰ − 1) + 6
Transcribed Image Text:By dragging statements from the left column to the right column below, give a proof by induction of the following statement:
an7" 1 is a solution to the recurrence relation an = 7an-1 +6 with a = 0.
The correct proof will use 8 of the statements below.
Statements to choose from:
Note that a = 7⁰ -1 1-1 = 0, as
required.
Let P(n) be the statement, "an = 7" – 1".
Now assume that P(k) is true for an arbitrary
integer k ≥ 1.
Then ak+1
Then 7k+1
=
7ak + 6, so P(k + 1) is true.
7a0 + 6.
−1= 7(7k – 1) + 6, which is true.
Note that a₁ =
Now assume that P(k + 1) is true.
This simplifies to
: 7k+¹ − 7 + 6 = 7k+1 _ 1
ak+1 =
Your Proof: Put chosen statements in order in this
column and press the Submit Answers button.
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