Lhe blahks Ih the föllowing proof, which shows that the sequence defined by the recurrence relation k = fk-1+ 2* for each integer k 22 atisfies the following formula. f. = 2" +1- 3 for every integer n21 "roof (by mathematical induction): uppose f, f, f... is a sequence that satisfies the recurrence relation f = f-1+ 2* for each integer k2 2, with initial condition f, = 1. Ve need to show that when the sequence f,, fa, fa,... is defined in this recursive way, all the terms in the seguence also satisfy the explicit formula shown above. So let the property P(n) be the equation f = 2" +1- 3. We will show that P(n) is true for every integer n 2 1. Show that P(1) is true: The left-hand side of P(1) is 4-3 which equals 1 . The right-hand side of P(1) is 1 Since the left-hand and right-hand sides equal each other, P(1) Show that for each integerkz 1, if P(k) is true, then P(k + 1) is true: Let k be any integer with k2 1, and suppose that P(k) is true. In other words, suppose that f, =ok+1_3 where f, f, f... is a sequence defined by the recurre f, = f, + 2* for each integer k 2 2, with initial condition f, = 1. [This is the inductive hypothesis.] We must show that P(k + 1) is true. In other words, we must show that f ,= k+2 3 Now the left-hand side of P(k + 1) is k+1 = + 2*+ 1 k+1-3 by definition of fs, fa, fs, .. + 2k +1 by inductive hypothesis %3D ok+1 = 2 ak+2 3 by the laws of algebra and this is the right-hand side of P(k + 1). Hence the inductive step is complete.
Lhe blahks Ih the föllowing proof, which shows that the sequence defined by the recurrence relation k = fk-1+ 2* for each integer k 22 atisfies the following formula. f. = 2" +1- 3 for every integer n21 "roof (by mathematical induction): uppose f, f, f... is a sequence that satisfies the recurrence relation f = f-1+ 2* for each integer k2 2, with initial condition f, = 1. Ve need to show that when the sequence f,, fa, fa,... is defined in this recursive way, all the terms in the seguence also satisfy the explicit formula shown above. So let the property P(n) be the equation f = 2" +1- 3. We will show that P(n) is true for every integer n 2 1. Show that P(1) is true: The left-hand side of P(1) is 4-3 which equals 1 . The right-hand side of P(1) is 1 Since the left-hand and right-hand sides equal each other, P(1) Show that for each integerkz 1, if P(k) is true, then P(k + 1) is true: Let k be any integer with k2 1, and suppose that P(k) is true. In other words, suppose that f, =ok+1_3 where f, f, f... is a sequence defined by the recurre f, = f, + 2* for each integer k 2 2, with initial condition f, = 1. [This is the inductive hypothesis.] We must show that P(k + 1) is true. In other words, we must show that f ,= k+2 3 Now the left-hand side of P(k + 1) is k+1 = + 2*+ 1 k+1-3 by definition of fs, fa, fs, .. + 2k +1 by inductive hypothesis %3D ok+1 = 2 ak+2 3 by the laws of algebra and this is the right-hand side of P(k + 1). Hence the inductive step is complete.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.6: Permutations
Problem 12E
Related questions
Question
For the box that I got it wrong. It is not looking for a number so what is the answer for it. Please help!
![Fill in the blanks in the following proof, which shows that the sequence defined by the recurrence relation
fk = fk -1
f = 1
+ 2k for each integer k 2 2
satisfies the following formula.
f. = 2" +1- 3 for every integer n 21
Proof (by mathematical induction):
Suppose f,, f,, fy... is a sequence that satisfies the recurrence relation
f, = f, + 2k for each integer k 2 2, with initial condition f, = 1.
We need to show that when the sequence f,, f, fay... is defined in this recursive way, all the terms in the sequence also satisfy the explicit formula shown above.
So let the property P(n) be the equation f = 2" + 1 - 3. We will show that P(n) is true for every integer n 2 1.
Show that P(1) is true:
The left-hand side of P(1) is 4-3
, which equals 1
. The right-hand side of P(1) is 1
Since the left-hand and right-hand sides equal each other, P(1) is true.
Show that for each integer k2 1, if P(k) is true, then P(k + 1) is true:
Let k be any integer with k 2 1, and suppose that P(k) is true. In other words, suppose that f, = 21-3
where f,, f,fa.. is a sequence defined by the recurrence relation
f = f,+ 2k for each integer k 2 2, with initial condition f, = 1.
[This is the inductive hypothesis.]
We must show that P(k + 1) is true. In other words, we must show that f1
k+2
3
. Now the left-hand side of P(k + 1) is
%3D
by definition of f1, f2, fs, ...
k+1 = f + 2k + 1
ok+1
3
+ 2k + 1
by inductive hypothesis
k+1
= 2
= ok+2 - 3
by the laws of algebra
and this is the right-hand side of P(k + 1). Hence the inductive step is complete.
[Thus, both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5515c3ba-7f5d-4ee4-b5cd-03b0cd5507cb%2F903b8976-5404-4ee0-bf00-7a3246f35884%2F2npu5sn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Fill in the blanks in the following proof, which shows that the sequence defined by the recurrence relation
fk = fk -1
f = 1
+ 2k for each integer k 2 2
satisfies the following formula.
f. = 2" +1- 3 for every integer n 21
Proof (by mathematical induction):
Suppose f,, f,, fy... is a sequence that satisfies the recurrence relation
f, = f, + 2k for each integer k 2 2, with initial condition f, = 1.
We need to show that when the sequence f,, f, fay... is defined in this recursive way, all the terms in the sequence also satisfy the explicit formula shown above.
So let the property P(n) be the equation f = 2" + 1 - 3. We will show that P(n) is true for every integer n 2 1.
Show that P(1) is true:
The left-hand side of P(1) is 4-3
, which equals 1
. The right-hand side of P(1) is 1
Since the left-hand and right-hand sides equal each other, P(1) is true.
Show that for each integer k2 1, if P(k) is true, then P(k + 1) is true:
Let k be any integer with k 2 1, and suppose that P(k) is true. In other words, suppose that f, = 21-3
where f,, f,fa.. is a sequence defined by the recurrence relation
f = f,+ 2k for each integer k 2 2, with initial condition f, = 1.
[This is the inductive hypothesis.]
We must show that P(k + 1) is true. In other words, we must show that f1
k+2
3
. Now the left-hand side of P(k + 1) is
%3D
by definition of f1, f2, fs, ...
k+1 = f + 2k + 1
ok+1
3
+ 2k + 1
by inductive hypothesis
k+1
= 2
= ok+2 - 3
by the laws of algebra
and this is the right-hand side of P(k + 1). Hence the inductive step is complete.
[Thus, both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning