Let a₁, a₂, 3, ... be the sequence defined recursively as follows. 3ak-1 + 2 for each integer k ≥ 1 Use iteration to guess an explicit formula for the sequence by filling in the blanks below. Simplify the result by using a formula from Section 5.2. 2₁ = 2 22 3a₁ + 2 = 3 1 23 = 35 332 +2 = 33. 2 = = 3 34 = 333 = 3 3 an = 3 = = 3 3 324 +2 = 3 ak a₁ = 2 = 3 = 2. = 2 + 2 = 3 = (3 n+1 v ??? ♥ 2 + 3 n+2 ✓ + 2 +2 3-1 +3 2 Based on this pattern, it is reasonable to guess that n+1 V n+2 ✓ n+3 by definition of a₁a₂a3--- +3. 1 +3. +3 n-1 V +3 +3 + 2 +2 2 + 3 + 2+3. . by definition of a ₁, ₂, 3 -- +3. +3' + ... + 3² + 3 + 1) + 2 + 2 by definition of a ₁, ₂, 3 --- + 2 + + 32.2 +3.2+2 +3. +2 + 2 by definition of a₁a₂a3--- [by Theorem 5.2.2 with r = 3] for every integer n ≥ 1.
Let a₁, a₂, 3, ... be the sequence defined recursively as follows. 3ak-1 + 2 for each integer k ≥ 1 Use iteration to guess an explicit formula for the sequence by filling in the blanks below. Simplify the result by using a formula from Section 5.2. 2₁ = 2 22 3a₁ + 2 = 3 1 23 = 35 332 +2 = 33. 2 = = 3 34 = 333 = 3 3 an = 3 = = 3 3 324 +2 = 3 ak a₁ = 2 = 3 = 2. = 2 + 2 = 3 = (3 n+1 v ??? ♥ 2 + 3 n+2 ✓ + 2 +2 3-1 +3 2 Based on this pattern, it is reasonable to guess that n+1 V n+2 ✓ n+3 by definition of a₁a₂a3--- +3. 1 +3. +3 n-1 V +3 +3 + 2 +2 2 + 3 + 2+3. . by definition of a ₁, ₂, 3 -- +3. +3' + ... + 3² + 3 + 1) + 2 + 2 by definition of a ₁, ₂, 3 --- + 2 + + 32.2 +3.2+2 +3. +2 + 2 by definition of a₁a₂a3--- [by Theorem 5.2.2 with r = 3] for every integer n ≥ 1.
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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I got lost in this problem. What is the pattern?
![Let a₁, a₂, 3, ... be the sequence defined recursively as follows.
3ak-1 + 2 for each integer k ≥ 1
Use iteration to guess an explicit formula for the sequence by filling in the blanks below. Simplify the result by using a
formula from Section 5.2.
2₁ = 2
22 3a₁ + 2
= 3 1
23
=
35
332 +2
= 33. 2
=
= 3
34 =
333
= 3 3
an
= 3
=
= 3 3
324 +2
= 3
ak
a₁ = 2
= 3
= 2.
= 2
+ 2
= 3
=
(3
n+1 v
??? ♥
2 + 3
n+2 ✓
+ 2
+2
3-1
+3
2
Based on this pattern, it is reasonable to guess that
n+1 V
n+2 ✓
n+3
by definition of a₁a₂a3---
+3.
1
+3.
+3
n-1 V
+3
+3
+ 2
+2
2 + 3
+
2+3.
.
by definition of a ₁, ₂, 3 --
+3.
+3'
+ ... + 3² + 3 + 1)
+ 2
+ 2
by definition of a ₁, ₂, 3 ---
+
2 + + 32.2 +3.2+2
+3.
+2
+ 2
by definition of a₁a₂a3---
[by Theorem 5.2.2 with r = 3]
for every integer n ≥ 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9824ebe1-81ec-4540-832f-9b2c74c8a4d5%2F27e7c9fd-35d5-43cf-b60b-84a6a5bb533f%2F00g7f9h_processed.png&w=3840&q=75)
Transcribed Image Text:Let a₁, a₂, 3, ... be the sequence defined recursively as follows.
3ak-1 + 2 for each integer k ≥ 1
Use iteration to guess an explicit formula for the sequence by filling in the blanks below. Simplify the result by using a
formula from Section 5.2.
2₁ = 2
22 3a₁ + 2
= 3 1
23
=
35
332 +2
= 33. 2
=
= 3
34 =
333
= 3 3
an
= 3
=
= 3 3
324 +2
= 3
ak
a₁ = 2
= 3
= 2.
= 2
+ 2
= 3
=
(3
n+1 v
??? ♥
2 + 3
n+2 ✓
+ 2
+2
3-1
+3
2
Based on this pattern, it is reasonable to guess that
n+1 V
n+2 ✓
n+3
by definition of a₁a₂a3---
+3.
1
+3.
+3
n-1 V
+3
+3
+ 2
+2
2 + 3
+
2+3.
.
by definition of a ₁, ₂, 3 --
+3.
+3'
+ ... + 3² + 3 + 1)
+ 2
+ 2
by definition of a ₁, ₂, 3 ---
+
2 + + 32.2 +3.2+2
+3.
+2
+ 2
by definition of a₁a₂a3---
[by Theorem 5.2.2 with r = 3]
for every integer n ≥ 1.
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