6.50 Prove that 241 (52h -1) for every positive integern. Proof: We proceed by induction. Let P(n): "241 (5²-1) VnEn. Since P(): "52-1=25-1=24" is divisible by 24; P(1) is true. Now let KEN assume P(K);"52h -1 = 24;" for jEZ, and show |P(K+1); " 5²/(1+1) - 1 = 24;." Now, observe that 2 (K+1) - 1 522.52 -1 2552-1 = 25 (24; +1)-1 = 25.24; +24 = 24 (25; +1) 10/30/22 This implies that 241 (52(x+1)-1). We [By P(x)] By the Principle of Mathematical Induction, 241 (520-1) for nEN. "/
6.50 Prove that 241 (52h -1) for every positive integern. Proof: We proceed by induction. Let P(n): "241 (5²-1) VnEn. Since P(): "52-1=25-1=24" is divisible by 24; P(1) is true. Now let KEN assume P(K);"52h -1 = 24;" for jEZ, and show |P(K+1); " 5²/(1+1) - 1 = 24;." Now, observe that 2 (K+1) - 1 522.52 -1 2552-1 = 25 (24; +1)-1 = 25.24; +24 = 24 (25; +1) 10/30/22 This implies that 241 (52(x+1)-1). We [By P(x)] By the Principle of Mathematical Induction, 241 (520-1) for nEN. "/
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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![10.50
Prove that 241 (520-1) for every positive integer n.
Proof: We procced by induction. Let P(n): "241 (5²-1)."
#ntn. Since P(); `` 5²0-1=25-1= 24" is divisible
by 24, P(1) is true. Now let KEN
assume P(K); " 52h -1 = 24;" for jtz, and show
$²2(6+¹) - | = 24;." Now, observe that
52(1+1)]
P(k+1); "
52k.52 -1
5
2 (K+1)
1
=
Tarin Jones
MS 300
Homework 11
10/30/22
25 52k-1
= 25 (24; +1)-1
= 25•24; +24
= 24 (25; +1)
that
=
We
241 (52(x+|)_ -1₂
[By P(K)]
This implies
By the Principle of Mathematical Induction,
241 (520-1)
for neN.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2cd0447-e2a9-4a27-b885-f1de1f9e9258%2F0bfa94bd-8141-4bf3-915e-facde2ef25fc%2F9wqbz2o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:10.50
Prove that 241 (520-1) for every positive integer n.
Proof: We procced by induction. Let P(n): "241 (5²-1)."
#ntn. Since P(); `` 5²0-1=25-1= 24" is divisible
by 24, P(1) is true. Now let KEN
assume P(K); " 52h -1 = 24;" for jtz, and show
$²2(6+¹) - | = 24;." Now, observe that
52(1+1)]
P(k+1); "
52k.52 -1
5
2 (K+1)
1
=
Tarin Jones
MS 300
Homework 11
10/30/22
25 52k-1
= 25 (24; +1)-1
= 25•24; +24
= 24 (25; +1)
that
=
We
241 (52(x+|)_ -1₂
[By P(K)]
This implies
By the Principle of Mathematical Induction,
241 (520-1)
for neN.
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