Show that 3 is a factor of n3 + 2n for all natural numbers n. Let P denote the statement: 3 is a factor of n3 + 2n for all natural numbers n. First, write a mathematical statement equivalent to Pn. To be a factor of n³ + 2n means that there is some natural number m such that n³ + 2n = Check that P₁ is true: Assume Pk is true: 3 + 3+2 +( = (k³+2k) + To show that Pk+1 is true, we must show that Expand the left-hand side, and then rewrite in the desired form. (k+ 1)3 + 2(k+ 1) = and 3= 3(1). Thus P₁ is true. +2k + 2 = 3m₁ for some natural number m₁. )³ + 2( = 3m2 for some natural number m2.
Show that 3 is a factor of n3 + 2n for all natural numbers n. Let P denote the statement: 3 is a factor of n3 + 2n for all natural numbers n. First, write a mathematical statement equivalent to Pn. To be a factor of n³ + 2n means that there is some natural number m such that n³ + 2n = Check that P₁ is true: Assume Pk is true: 3 + 3+2 +( = (k³+2k) + To show that Pk+1 is true, we must show that Expand the left-hand side, and then rewrite in the desired form. (k+ 1)3 + 2(k+ 1) = and 3= 3(1). Thus P₁ is true. +2k + 2 = 3m₁ for some natural number m₁. )³ + 2( = 3m2 for some natural number m2.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Since m₁ +
true for all natural numbers n.
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11
||
(k³ + 2k) +
3(
Watch
m₁ +31
1 (m₂ + (
m1
)
))
is a natural number, Pk+1 is true. We conclude by the principle of mathematical induction that Pn is
![个
↑
D
Show that 3 is a factor of n3 + 2n for all natural numbers n.
Let P denote the statement: 3 is a factor of n3 + 2n for all natural numbers n.
First, write a mathematical statement equivalent to Pn. To be a factor of n³ + 2n means that there is some natural number m such
that n³ + 2n =
Check that P₁ is true:
Assume Pk is true:
3
(k + 1)³ + 2(k + 1) =
2(
+21
3+2
])=
= (k³ + 2k) +
=
=
+ 2k + 2
and 3 = 3(1). Thus P₁ is true.
To show that Pk+1 is true, we must show that
Expand the left-hand side, and then rewrite in the desired form.
3m₁ for some natural number m₁.
3
)³ + ²(
2
= 3m₂ for some natural number m2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9f36f111-f1bc-4b8c-9508-d8f50a3e95e1%2F8f4d624e-5b54-4a30-a0d1-465c8f6543ad%2Ft58adtx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:个
↑
D
Show that 3 is a factor of n3 + 2n for all natural numbers n.
Let P denote the statement: 3 is a factor of n3 + 2n for all natural numbers n.
First, write a mathematical statement equivalent to Pn. To be a factor of n³ + 2n means that there is some natural number m such
that n³ + 2n =
Check that P₁ is true:
Assume Pk is true:
3
(k + 1)³ + 2(k + 1) =
2(
+21
3+2
])=
= (k³ + 2k) +
=
=
+ 2k + 2
and 3 = 3(1). Thus P₁ is true.
To show that Pk+1 is true, we must show that
Expand the left-hand side, and then rewrite in the desired form.
3m₁ for some natural number m₁.
3
)³ + ²(
2
= 3m₂ for some natural number m2.
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