Let p1, p2, p3 … be a sequence defined recursively as follows. pk = pk − 1 + 2 · 3k for each integer k ≥ 2 p1 = 2 Use mathematical induction to show that such a sequence satisfies the equation pn = 3n + 1 − 7 for every integer n ≥ 1. Need help with proofing.
Let p1, p2, p3 … be a sequence defined recursively as follows. pk = pk − 1 + 2 · 3k for each integer k ≥ 2 p1 = 2 Use mathematical induction to show that such a sequence satisfies the equation pn = 3n + 1 − 7 for every integer n ≥ 1. Need help with proofing.
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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Let p1, p2, p3 … be a sequence defined recursively as follows.
| pk | = | pk − 1 + 2 · 3k for each integer k ≥ 2 |
| p1 | = | 2 |
Use mathematical induction to show that such a sequence satisfies the equation pn = 3n + 1 − 7 for every integer n ≥ 1.
Need help with proofing.
![The selected statement is true because the two sides of the equation equal each other.
Show that for every integer k ≥ 1, if P(k) is true then P(k + 1) is true:
Let k be any integer with k ≥ 1, and suppose that P(k) is true, where P(k) is the equation
Pk= 3+1
X
This is the inductive hypothesis
=
We must show that P(k+ 1) is true, where P(k+ 1) is the equation Pk +1
The left-hand side of P(k+ 1) is
? V
Pk + 1
=
=
P₁
3
= 3
?
?
= 3.3
?
= 3
V
?
V
+2.3
- 7.
V
(1+2) - 7
+2.3
- 7
7
?
by the laws of algebra,
which is the right-hand side of P(k+ 1) [as was to be shown].
by definition of P₁ P₂ P3¹ ---
by inductive hypothesis
- 7.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9824ebe1-81ec-4540-832f-9b2c74c8a4d5%2F4968c33d-7f50-4e06-bde7-23eb9f6d2f5b%2Ftgv7my_processed.png&w=3840&q=75)
Transcribed Image Text:The selected statement is true because the two sides of the equation equal each other.
Show that for every integer k ≥ 1, if P(k) is true then P(k + 1) is true:
Let k be any integer with k ≥ 1, and suppose that P(k) is true, where P(k) is the equation
Pk= 3+1
X
This is the inductive hypothesis
=
We must show that P(k+ 1) is true, where P(k+ 1) is the equation Pk +1
The left-hand side of P(k+ 1) is
? V
Pk + 1
=
=
P₁
3
= 3
?
?
= 3.3
?
= 3
V
?
V
+2.3
- 7.
V
(1+2) - 7
+2.3
- 7
7
?
by the laws of algebra,
which is the right-hand side of P(k+ 1) [as was to be shown].
by definition of P₁ P₂ P3¹ ---
by inductive hypothesis
- 7.
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