The right-hand side of P(K) is k. 2" +2 [The inductive hypothesis states that the two sides of P(k) are equal. i - 2' • (k+2)2(k+2) The right-hand side of P(k + 1) is k+2n+2) We must show that P(k + 1) is true. The left-hand side of P(k + 1) is i- 2'. when the final term of S i.2' is written separately, the result is i-2 (24 (n-2) (k + 2)2* + 2). After substitution from the inductive hypothesis, the left-hand side of P(k + 1) becomes When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal + 2. Hence P(k + 1) is true, which completes the inductive step. [Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]
The right-hand side of P(K) is k. 2" +2 [The inductive hypothesis states that the two sides of P(k) are equal. i - 2' • (k+2)2(k+2) The right-hand side of P(k + 1) is k+2n+2) We must show that P(k + 1) is true. The left-hand side of P(k + 1) is i- 2'. when the final term of S i.2' is written separately, the result is i-2 (24 (n-2) (k + 2)2* + 2). After substitution from the inductive hypothesis, the left-hand side of P(k + 1) becomes When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal + 2. Hence P(k + 1) is true, which completes the inductive step. [Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]
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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Please read the question and leave notes on the answer where appropriate. Please DO NOT skip any steps. Please double check your answer. DO NOT SUBMIT A TYPED RESPONSE. Please use a hand written response that is LEGABLE. Please review your answer to make sure all steps are clearly visable. Please double check your work. Thank you.
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![The right-hand side of P(k) is k. k+2)
+ 2
[The inductive hypothesis states that the two sides of P(k) are equal.
(k+1)+1
i· 2' = i. 2' + (k+2)2(k+2)
=1
(k+1)+1
(k+1)+1
k+1
Σ
We must show that P(k + 1) is true. The left-hand side of P(k + 1) is
i. 2', when the final term of
i. 2' is written separately, the result is
· The right-hand side of P(k + 1) isk+2n+2)
i = 1
i = 1
i = 1
2(n+2)
+(k + 2)2k
+ 2. Hence P(k + 1)
After substitution from the inductive hypothesis, the left-hand side of P(k + 1) becomes
When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal
is true, which completes the inductive step.
[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%2Fafa4cb6f-0c50-4ba8-9a03-e3918106ad13%2F5848465f-c932-4d45-b1b7-4fe8442d6e04%2Fcnw03as_processed.png&w=3840&q=75)
Transcribed Image Text:The right-hand side of P(k) is k. k+2)
+ 2
[The inductive hypothesis states that the two sides of P(k) are equal.
(k+1)+1
i· 2' = i. 2' + (k+2)2(k+2)
=1
(k+1)+1
(k+1)+1
k+1
Σ
We must show that P(k + 1) is true. The left-hand side of P(k + 1) is
i. 2', when the final term of
i. 2' is written separately, the result is
· The right-hand side of P(k + 1) isk+2n+2)
i = 1
i = 1
i = 1
2(n+2)
+(k + 2)2k
+ 2. Hence P(k + 1)
After substitution from the inductive hypothesis, the left-hand side of P(k + 1) becomes
When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal
is true, which completes the inductive step.
[Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]
Transcribed Image Text:Prove the following statement by mathematical induction.
n +1
For every integer n 2 0, Fi. 2' = n. 2" + 2 + 2.
i = 1
Proof (by mathematical induction): Let P(n) be the equation
i. 2' = n. 2" + 2 + 2.
We will show that P(n) is true for every integer n 2 0.
Show that P(0) is true: Select P(0) from the choices below.
0 +1
S 1.2' = 1 . 21 + 2 +
2 = 0- 2" + 2 + 2
0 +1
Si. 2' = 0. 2º + 2 + 2
i = 1
n + 1
oFi.2' = 0 · 2º + 2 + 2
%3D
i = 1
The selected statement is true because both sides of the equation equal the same quantity.
Show that for each integer k > 0, if P(k) is true, then P(k + 1) is true:
Let k be any integer with k 2 0, and suppose that P(k) is true. Select the expression for the left-hand side of P(k) from the choices below.
k+1
k· 2
i = 1
k+1
Ei. 2
i = 1
k+1
i = 1
k+1
i-2k +
i = 1
1
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