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By dragging statements from the left column to the right column below, give a proof by induction of the following statement: 9"-1 is a solution to the recurrence relation a, 9a-1 +8 with ag = 0. a,

By dragging statements from the left column to the right column below, give a proof by induction of the following statement: 9"-1 is a solution to the recurrence relation a, 9a-1 +8 with ag = 0. a,

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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Note that a = 90-1=1-1=0, as required. Note that a₁ = 9a0 +8. Now assume that P(n) is true for all n ≥ 0. Then 9+1 -1 = 9(9k-1)+8, which is true. This simplifies to ak+1 = 9+19+8 = 9+1 - 1 Now assume that P(k) is true for an arbitrary integer 1. Let P(n) be the statement, "a,,= 9" - 1 is a solution to the recurrence relation an = 9an-1 +8 with a = 0." Thus P(k+ 1) is true. By the recurrence relation, we have ak+1 = 9ak + 8 = 9(9k-1) + 8 Thus P(k) is true for all k.
Transcribed Image Text:Note that a = 90-1=1-1=0, as required. Note that a₁ = 9a0 +8. Now assume that P(n) is true for all n ≥ 0. Then 9+1 -1 = 9(9k-1)+8, which is true. This simplifies to ak+1 = 9+19+8 = 9+1 - 1 Now assume that P(k) is true for an arbitrary integer 1. Let P(n) be the statement, "a,,= 9" - 1 is a solution to the recurrence relation an = 9an-1 +8 with a = 0." Thus P(k+ 1) is true. By the recurrence relation, we have ak+1 = 9ak + 8 = 9(9k-1) + 8 Thus P(k) is true for all k.
By dragging statements from the left column to the right column below, give a proof by induction of the following statement: The correct proof will use 8 of the statements below. Statements to choose from: Let P(n) be the statement, "a, = 9"-1". Therefore, by the Principle of Mathematical Induction, P(n) ise for all n ≥ 1. Then ak+1=9ak + 8, so P(k+ 1) is true. Then = 9 - 1. Now assume that P(k+ 1) is true. Note that a = 90-1-1-1=0, as required. Note that a₁ = 9ao + 8. Now assume that P(n) is true for all n ≥ 0. Then 9+1 -1 = 9(91)+8, which is true. an = 9"-1 is a solution to the recurrence relation a, 9a-1 +8 with ag = 0. Your Proof: Put chosen statements in order in this column and press the Submit Answers button.
Transcribed Image Text:By dragging statements from the left column to the right column below, give a proof by induction of the following statement: The correct proof will use 8 of the statements below. Statements to choose from: Let P(n) be the statement, "a, = 9"-1". Therefore, by the Principle of Mathematical Induction, P(n) ise for all n ≥ 1. Then ak+1=9ak + 8, so P(k+ 1) is true. Then = 9 - 1. Now assume that P(k+ 1) is true. Note that a = 90-1-1-1=0, as required. Note that a₁ = 9ao + 8. Now assume that P(n) is true for all n ≥ 0. Then 9+1 -1 = 9(91)+8, which is true. an = 9"-1 is a solution to the recurrence relation a, 9a-1 +8 with ag = 0. Your Proof: Put chosen statements in order in this column and press the Submit Answers button.
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